A Simplified Approach to Multi-carrier Modulation

A SIMPLIFIED APPROACH TO MULTI-CARRIER MODULATION

A Thesis

Submitted to the Faculty

of

Purdue University

by

Andrew C. Marcum

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Engineering

May 2010

Purdue University

Fort Wayne, Indiana

ii

For my grandfather Alan and my grandmothers Dorothy and Rita.

iii

ACKNOWLEDGMENTS

I first thank Dr. Todor Cooklev for the support and leadership he provided me

throughout the last two semesters. I am very thankful for the opportunity to work under

the guidance and teachings of Dr. Cooklev, and I am better engineer for it. Next, I thank

my graduate committee; Dr. Steven Walter, Dr. Carlos Raez and Dr. Tim Grove and

thesis format director, Barbara Lloyd for the time and effort expended on my behalf. I

thank Raytheon Company for supporting my efforts and desires to further my education.

At times, managing the requirements of graduate school in conjunction with a demanding

job can be very stressful and difficult to balance. As such, I am thankful to work for a

company that fosters an environment where education is valued and the goals of its

employees are supported. I thank my family for supporting my dreams and providing me

every possible opportunity to reach this milestone. Last but not least, I thank my fiancé,

Rebecca. Rebecca sacrificed a lot to be with me and has been nothing but supportive

during the many hours I have put into graduate school above and beyond the duties of

work.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES ........................................................................................................................... v

LIST OF FIGURES ........................................................................................................................ vi

LIST OF ABBREVIATIONS ......................................................................................................... ix

ABSTRACT ..................................................................................................................................... x

1. INTRODUCTION ...................................................................................................................... 1

2. CONVENTIONAL SYSTEM .................................................................................................... 3

2.1 SISO System Description .............................................................................................. 3

2.2 SISO OFDM Description ............................................................................................... 5

2.3 MIMO System Description .......................................................................................... 10

2.4 V-Blast MIMO OFDM Description ............................................................................. 14

3. SIMPLIFIED SYSTEM ............................................................................................................ 26

3.1 Simple System Description ........................................................................................... 26

3.2 Simple Discrete Fourier Transform Matrix .................................................................. 27

3.3 Simple Inverse Discrete Fourier Transform Matrix ...................................................... 32

3.4 Simple Fast Fourier Transform Algorithm ................................................................... 34

3.5 Simple Inverse Fast Fourier Transform Algorithm ....................................................... 38

3.6 Simple SISO OFDM ..................................................................................................... 43

3.7 Simple MIMO OFDM .................................................................................................. 47

4. SIMULATION RESULTS ....................................................................................................... 54

4.1 SISO OFDM Architecture ............................................................................................ 54

4.2 MIMO OFDM Architecture .......................................................................................... 70

5. CONCLUSIONS....................................................................................................................... 79

BIBLIOGRAPHY .......................................................................................................................... 83

v

LIST OF TABLES

Table Page

4.1 Rounded FFT/IFFT Twiddle Factor Quantization ......................................................54

5.1 FFT and Rounded FFT Complexity ............................................................................80

5.2 Complexity Reduction Provided by Rounded FFT ....................................................81

vi

LIST OF FIGURES

Figure Page

2.1. SISO OFDM System....................................................................................................4

2.2. SISO OFDM Transceiver Block Diagram ...................................................................5

2.3. MIMO OFDM System ...............................................................................................11

2.4. SISO Capacity vs. MIMO Capacity...........................................................................13

2.5. V-Blast MIMO OFDM Receiver/Transmitter Block Diagram ..................................14

3.1. Response of Five Level Rounded Sine (k=2) ............................................................29

3.2. Response of Seventeen Level Rounded Sine (k=8) ...................................................30

3.3. Four Point Radix-4 FFT Butterfly Diagram ..............................................................36

3.4. Four Point Rounded Radix-4 FFT Butterfly Diagram ...............................................38

3.5. Four Point Radix-4 IFFT Butterfly Diagram .............................................................40

3.6. Four Point Rounded Radix-4 IFFT Butterfly Diagram..............................................43

3.7. Simplified SISO OFDM Transceiver Block Diagram ...............................................44

3.8. Simplified V-Blast MIMO OFDM Receiver/Transmitter Block Diagram ................48

4.1. Flat Fading Channel Frequency Response (Channel 1) .............................................55

4.2. Typical Office Channel Frequency Response (Channel 2)........................................56

4.3. Large Open Area Channel Frequency Response (Channel 3) ...................................57

4.4. SISO OFDM with QPSK BER, k=2, Channel 1 ........................................................58

vii

Figure Page



4.7. SISO OFDM with QPSK BER, k=16, Channel 1 ......................................................59

4.8. SISO OFDM with 16QAM BER, k=2, Channel 1 ....................................................60

4.9. SISO OFDM with 16QAM BER, k=4, Channel 1 ....................................................60

4.10. SISO OFDM with 16QAM BER, k=8, Channel 1 ..................................................61

4.11. SISO OFDM with 16QAM BER, k=16, Channel 1 ................................................61




4.15. SISO OFDM with QPSK BER, k=16, Channel 2 ....................................................63








4.23. SISO OFDM with QPSK BER, k=16, Channel 3 ....................................................67




viii

Figure Page


4.28. MIMO OFDM with Optimal Ordered ZF-SIC and QPSK BER, k=2 .....................71



4.31. MIMO OFDM with Optimal Ordered ZF-SIC and QPSK BER, k=16 ...................72

4.32. MIMO OFDM with Optimal Ordered MMSE-SIC and QPSK BER, k=2 ..............73



4.35. MIMO OFDM with Optimal Ordered MMSE-SIC and QPSK BER, k=16 ............74

4.36. MIMO OFDM with Optimal Ordered ZF-SIC and 16QAM BER, k=2 ..................75



4.39. MIMO OFDM with Optimal Ordered ZF-SIC and 16QAM BER, k=16 ................76

4.40. MIMO OFDM with Optimal Ordered MMSE-SIC and 16QAM BER, k=2 ...........77



4.43. MIMO OFDM with Optimal Ordered MMSE-SIC and 16QAM BER, k=16 .........78

ix

LIST OF ABBREVIATIONS

AWGN Additive White Gaussian Noise

BER Bit Error Rate

CP Cyclic Prefix

DFT Discrete Fourier Transform

FFT Fast Fourier Transform

ICI Inter-Carrier Interference

IDFT Inverse Discrete Fourier Transform

IFFT Inverse Fast Fourier Transform

ISI Inter-Symbol Interference

MIMO Multiple Input, Multiple Output

MMSE Minimum Mean Square Error

MRC Maximal Ratio Combining

OFDM Orthogonal Frequency Division Multiplexing

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

SIC Successive Interference Cancellation

SISO Single Input, Single Output

SWaP Size Weight and Power

V-Blast Vertical-Bell Laboratories-Layered-Space-Time

ZF Zero Forcing

x

ABSTRACT

Marcum, Andrew C. M.S.E., Purdue University, May 2010. A Simplified Approach to

Multi-Carrier Modulation. Major Professor: Steven Walter.

There is a significant demand for a decrease in the size, weight and power

(SWaP) associated with wireless systems. In recent years, multiple-input, multiple-

output (MIMO) wireless systems have received considerable attention due to the high

data rates they provide. Orthogonal frequency division multiplexing (OFDM), a digital

multi-carrier modulation technique, is well suited to be used in MIMO systems as it

provides the ability to operate in frequency-selective channel environments. When

OFDM is combined with the capacity increase provided by MIMO systems, the result is a

very successful communication system. In this research, a reduced-complexity MIMO

OFDM system is advanced. The proposed system is multiplier-less and thus requires a

simpler digital hardware implementation. As a result, the chip area, power consumption

and cost associated with the MIMO OFDM system can be significantly reduced.

The reduction in complexity is obtained via modification to conventional Fast

Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) algorithms

necessary to implement OFDM multi-carrier modulation. System computational

complexity is reduced by quantizing what are known as “twiddle factors” in traditional

xi

FFT algorithms such as the Radix-2 and Radix-4. The quantization allows for all

multiplications to be done with a value of one, negative one, zero or a power of two.

Ensuring that all multiplications are performed with any of the aforementioned values

results in a transform where all multiplications are considered trivial. Replacing standard

multiplications with trivial multiplications significantly reduces system computational

complexity. As an example, the complexity associated with the implementation of the

rounded FFT as compared to a conventional Radix-4 FFT is reduced by 47% when

numerical values are represented with 16 bits. Depending on the application, different

quantization levels can be utilized in order to obtain the necessary performance

characteristics. As the number of quantization levels grows, the system capability

increasingly approaches the performance of a system that uses the conventional

transforms. When applied to MIMO OFDM systems, the computational savings are

significant as the combination of the IFFT and FFT algorithms are implemented for every

spatial stream (i.e. antenna). As such, the simplified approach provides a system that is a

lower-cost, practical alternative to the MIMO OFDM systems used today.

1

1. INTRODUCTION

The motivation for this research is to determine a solution that allows for a

reduction of computational complexity when applied to the implementations of existing

wireless communication technologies. Specifically, an investigation is performed to

simplify both single-input, single-output (SISO) and multiple-input, multiple-output

(MIMO) multi-carrier modulation systems with Orthogonal Frequency Division

Multiplexing (OFDM). One such method to reduce the complexity associated with the

implementation of SISO and MIMO systems is to simplify the processing of the Fast

Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) as required by

OFDM modulation. As with every simplification, there is a trade associated with system

performance that must occur. In this context, inclusion of the simplified FFT and IFFT

algorithms will result in some reduction of overall system performance when compared

to conventional systems, with the critical parameter affected by the simplification being

bit error rate (BER). In order to assess the impact, this analysis describes the

performance delta between the conventional system and the simplified system derived

from a comprehensive set of computer simulations [1]. The simulations model several

different system configurations, including multiple OFDM bit-to-symbol mapping

techniques. Utilizing an approach that considers many different implementations permits

this research to identify the schemes that provide optimal performance. Furthermore, the

2

results of the computer simulations, when coupled with the analysis of computational

complexity, provide the critical information necessary to determine whether or not the

simplified design can be considered a practical and viable communication system.

3

2. CONVENTIONAL SYSTEM

2.1 SISO System Description

The first system considered in this research utilizes an architecture that consists of

a single transmitter and single receiver, known as SISO. The modulation technique

utilized in conjunction with the SISO architecture is OFDM. OFDM can be characterized

as a digital modulation scheme that multiplexes complex data symbols and transmits the

symbols on multiple carriers that are closely spaced in frequency and orthogonal to one

another (considered a single OFDM symbol) [2, 3, 4]. In this configuration of multi-

carrier modulation, orthogonality between the closely spaced carriers is essential in order

to eliminate crosstalk and cancellation otherwise known as inter-carrier interference (ICI)

[2, 3]. Conventional bit-to-symbol mappings such as Quadrature Phase Shift Keying

(QPSK) and Quadrature Amplitude Modulation (QAM) are employed to obtain the

information transmitted by each carrier. In OFDM modulation, symbols are transmitted

by the carriers at a low rate, thus simplifying the hardware implementation of both the

transmitter and receiver. The combination of the set of low symbol rate carriers (OFDM

symbol) transmitted and received in parallel results in high a data rate system of modest

complexity.

As previously stated, orthogonality between carriers must be ensured in order to

realize the advantages of OFDM modulation. It is well known that the result of

4

computing the Fourier Transform with respect to a set of complex values results in an

orthogonal data set. As such, standard configurations of OFDM utilize the IFFT

algorithm in the transmitter and corresponding FFT in the receiver [1, 2, 3]. The IFFT is

performed for each carrier transmission and provides a time domain representation of the

complex symbols generated as a result of bit-to-symbol mapping schemes. The FFT

algorithm is utilized in the receiver in order to reverse the effects of the IFFT that is

implemented in the transmitter by converting the data into a frequency domain

representation. The frequency domain depiction of the data contains the original

complex symbol data with additional effects of the channel and noise. Because the

symbols are represented in the frequency domain, the removal of the channel

characteristics, known as equalization, is simplified as channel de-convolution can be

implemented by dividing the channel frequency response from each carrier. This method

of equalization is commonly referred to as zero-forcing (ZF) as the original data (with

additive noise) can be obtained simply through one division per OFDM carrier. To

further describe the explanation of SISO OFDM, Figure 2.1 is provided to illustrate the

system.

IFFT P/S

ChannelX(0)

X(1)

X(N-1)

.

.

.

.

.

.

S/P FFT.

.

.

Equalizer.

.

.

.

.

.

Y(0)

Y(1)

Y(N-1)

Fig. 2.1. SISO OFDM System

5

Another advantage of OFDM modulation is its ability to operate in frequency

selective channel conditions that can be harmful to the reliability of high data rate single-

carrier systems. Because of the slow data rate and thus small bandwidth associated with

each carrier, OFDM modulation can operate successfully in frequency selective

environments as the channel response can be considered flat with respect to a specific

carrier. In general, OFDM can be viewed as a set of slowly-modulated narrowband

signals as opposed to one rapidly modulated wideband signal associated with single-

carrier systems.

2.2 SISO OFDM Description

To begin a detailed discussion of the SISO system, consider an OFDM multi-

carrier modulation system with single transmit and receive antennas, as illustrated in

Figure 2.2.

IFFT P/S

X(0)

X(1)

X(N-1)

.

.

.

S/P FFT

Channel

Estimation

.

.

.

.

.

.

Y(0)

Y(1)

Y(N-1)

Add

CP

.

.

.

h +

w

Remove

CP

.

.

.

FFT

\

.

.

.

\

\

.

.

.

Fig. 2.2. SISO OFDM Transceiver Block Diagram

6

It can be quickly observed that the system represented in Figure 2.2 seems significantly

more complex than that in Figure 2.1. The added complexity is due to the fact that

Figure 2.2 includes a reference to the cyclic prefix (CP), channel impulse response model

as well as an expanded view of equalization. The complex information symbols are

denoted in Figure 2.2 by 𝑋(𝑖), where 𝑖 = 0,1, … , 𝑁 − 1 and 𝑁 is the total number of

carriers. The values of the complex symbols are derived from bit-to-symbol mapping

techniques such as M-ary Quadrature Amplitude Modulation (QAM). The IFFT block

represented in Figure 2.2 provides the capability to transform complex information

symbols, represented by 𝑿, into a time domain representation via a standard algorithm

[2]. Execution of the IFFT algorithm ensures carrier orthogonality during transmission,

which is a necessary requirement to successfully implement OFDM communications. In

this particular description, the IFFT length is equal to the number of carriers associated

with 𝑿, defined as 𝑁. In order to provide a mathematical representation of the IFFT, the

notation 𝑭𝑵−1 is introduced in Equation 2.1 to represent the Inverse Discrete Fourier

Transform (IDFT) matrix of size 𝑁𝑥𝑁.

𝒙 = 𝑭𝑵−1𝑿 (2.1)

Vector 𝒙 is the result of performing the IDFT, which is the length 𝑁 time domain

representation of 𝑿. Proceeding through Figure 2.2, a cyclic prefix (CP) of length 𝐾 is

applied to vector 𝒙. Inclusion of the CP results in a data packet with length 𝑁 + 𝐾. The

CP is a necessary component of OFDM modulation as it prevents inter-symbol

interference (ISI) that occurs as a result of multi-path. The process of applying the CP to

vector 𝒙 is described in Equation 2.2.

7

𝒙𝒄𝒑 = 𝑥 𝑁 − 𝐾 , 𝑥 𝑁 − 𝐾 − 1 , …𝑥 𝑁 − 1 , 𝑥 0 , 𝑥 1 , … , 𝑥 𝑁 − 1 (2.2)

In order to determine the appropriate CP length 𝐾, the multi-path components of the

wireless channel must be understood. The wireless channel can be modeled

mathematically as a finite impulse response (FIR) transfer function with 𝐿 taps or channel

coefficients. The wireless channel FIR transfer function is defined in Equation 2.3.

𝐻 𝑧 = 𝑕0 + 𝑕1𝑧−1 + ⋯ + 𝑕𝐿−1𝑧

−𝐿+1 (2.3)

In this analysis, it is assumed that the channel can be characterized by slow fading

and thus the channel impulse response does not change within one OFDM symbol. In

order for the CP to be effective in eliminating the effects of ISI, the CP length must

exceed the duration of the channel impulse response or more specifically, the number of

multi-path channel components as defined by Equation 2.4.

𝐾 ≥ 𝐿 − 1 (2.4)

Once the CP is incorporated into 𝒙, 𝒙𝒄𝒑 is transmitted through the wireless channel. At

the receiver, signal 𝒚𝒄𝒑 can be mathematically represented by the linear convolution

between transmitted signal 𝒙𝒄𝒑 and the channel impulse response. Channel noise 𝒘 is

also added to the received signal as specified in Equation 2.5 and Equation 2.6.

𝒚𝒄𝒑 = 𝒙𝒄𝒑 ∗ 𝒉𝒍 + 𝒘 (2.5)

𝑦𝑐𝑝 𝑚 = 𝑕𝑙𝑥𝑐𝑝 𝑚 − 𝑙 + 𝑤 𝑚 , 𝑚 = 0,1, …𝑁 + 𝐾 + 𝐿 − 2𝐿−1𝑙=0 (2.6)

Equation 2.5 generically describes the linear convolution, whereas Equation 2.6

characterizes the linear convolution by its mathematical definition. Channel noise 𝒘,

referenced in both Equation 2.5 and Equation 2.6, is Additive White Gaussian Noise

(AWGN) with zero mean and variance 𝜎2 =𝑁0

2, where 𝑁0 is the single-sided power

8

spectral density [1]. In OFDM modulated systems, the assumption is made that ISI

occurs for the first 𝐾 received symbols (received CP symbols) and thus these symbols

contained in received signal 𝒚𝒄𝒑 are discarded by the receiver. The process of removing

the CP to define signal 𝒚 is specified in Equation 2.7.

𝒚 = 𝒚𝒄𝒑 𝐾 : (𝑁 + 𝐾 − 1) (2.7)

The removal of the CP eliminates ISI, but there is another observation that can be made.

Removal of the CP converts the linear convolution between the transmission and the

channel impulse response as defined in Equation 2.6, into a cyclic convolution. To

further explore the observation of the cyclic convolution, consider representing the

channel in the format as indicated by Equation 2.8 and received signal 𝒚 with cyclic

prefix removed as indicated by Equation 2.9, where 𝑇 ∗ is the conjugate transpose

operation.

𝒉 = [𝑕0 , 𝑕1 , … , 𝑕𝐿−1, 0, … 0]𝑇∗ (2.8)

𝒚 = [𝑦 𝑘 , 𝑦 𝑘 + 1 , … , 𝑦[𝑁 + 𝐾 − 1]]𝑇∗ (2.9)

In order to mathematically describe the cyclic convolution of channel 𝒉 and 𝒙, a cyclic

matrix representation of channel 𝒉, notated as 𝒉 is defined in Equation 2.10.

𝒉 =

0 1 2 1

1 0 1 2

1 2 1 0

0 ... 0 ...

0 ... 0 ...

...

0 ... 0 ...

L L

L

L L

h h h h

h h h h

h h h h

H

(2.10)

Therefore, the received signal 𝒚 can be expressed as follows.

𝒚 = 𝒉 𝒙 + 𝒘 (2.11)

9

Through consideration of the relationship defined by Equation 2.1, signal 𝒚 can further

be specified as indicated in Equation 2.12.

𝒚 = 𝒉 𝑭𝑵−𝟏𝑿 + 𝒘 (2.12)

The next step in the receive chain of the SISO system is to apply the Fast Fourier

Transform (FFT) with respect to received signal 𝒚. Similar to the system’s definition of

the IFFT algorithm in the transmitter, the FFT length is equal to 𝑁. In order to provide a

mathematical representation of the FFT, notation 𝑭𝑵 is introduced as in Equation 2.13 in

order to represent the Discrete Fourier Transform (DFT) matrix of size 𝑁𝑥𝑁.

𝒀 = 𝑭𝑵𝒚 (2.13)

Furthermore, Equation 2.13 can be substituted into Equation 2.12 in order to define the

relationship presented in Equation 2.14.

𝒀 = 𝑭𝑵𝒉 𝑭𝑵−𝟏𝑿 + 𝑭𝑵𝒘 (2.14)

In order for the equations defined in this chapter to successfully represent a

communication system, the complex information symbols that originated as 𝑿 must be

recovered in some manner from 𝒀. The process of recovering the original symbols from

the received signal is known as equalization. Equalization can be accomplished in SISO

OFDM by multiplying the inverse of relationship 𝑭𝑵𝒉 𝑭𝑵−𝟏 with 𝒀, as noted in Equation

2.15.

𝑿 = 𝑭𝑵𝒉 𝑭𝑵−𝟏

−1𝒀 (2.15)

Typically, matrix inversion is extremely process intensive and in many cases, can only be

approximated; however, matrix inversion of 𝑭𝑵𝒉 𝑭𝑵−𝟏 is greatly simplified numerically

due to the fact that DFT multiplication diagonalizes circular matrices.

10

𝑭𝑵𝒉 𝑭 𝑵 = 𝐻[1] 0 0

0 ⋱ 00 0 𝐻[𝑁 − 1]

(2.16)

In Equation 2.16, 𝑯 represents the 𝑁 - point DFT of the channel impulse response 𝒉. As

such, it can be observed that the diagonalization effectively decomposes the channel into

parallel, ISI-free sub-channels. In other words, the frequency-selective channel is

transformed into a channel with flat fading per carrier. From the perspective of the

physical implementation, an estimate of 𝑿 can also be obtained with a simple zero-

forcing (ZF) detector that requires one division per carrier as defined in Equation 2.17.

𝑋 𝑀 = 𝑌 𝑀

𝐻 𝑀 𝑤𝑕𝑒𝑟𝑒 𝑀 = 0,1, … , 𝑁 − 1 (2.17)

Once an estimate of 𝑿 is determined (noted as 𝑿 ), the probability of bit error 𝑃𝑒 or BER,

is computed in order to measure the communication system performance. As an

example, for a 𝑁 carrier OFDM system with cyclic prefix, QAM bit-to-symbol mapping

scheme, AWGN channel and theoretical probability of QAM bit error 𝑃𝑄𝐴𝑀 𝐸𝑏

𝑁0 where

𝐸𝑏 is the energy per bit, the BER is defined by Equation 2.18.

𝑃𝑒 =1

𝑁 𝑃𝑄𝐴𝑀

𝐻[𝑘] 2𝑁𝐸𝑏

𝑁+𝐾 𝑁0 𝑁−1

𝑘=0 (2.18)

2.3 MIMO System Description

The second system considered in this research utilizes an architecture that consists

of multiple transmitters and multiple receivers, known as MIMO. In this particular

system, OFDM modulation is supported by each transmit and receive chain in a manner

similar to the system introduced for the SISO architecture. OFDM modulation is utilized

11

in conjunction with MIMO to obtain all of the benefits OFDM provides for a

communication system. To start, consider the MIMO OFDM system shown in Figure

2.3.

IFFT P/S

Channel

X1(0)

X1(1)

X1(N-1)

.

.

.

.

.

.

S/P FFT.

.

.

V-Blast

Symbol

Detection

.

.

.

.

.

.

Y1(0)

Y1(1)

Y1(N-1)

IFFT P/S

Xm(0)

Xm(1)

Xm(N-1)

.

.

.

.

.

.

S/P FFT.

.

.

.

.

.

.

.

.

Yn(0)

Yn(1)

Yn(N-1)

Fig. 2.3. MIMO OFDM System

When assessing the description of the MIMO OFDM system contained in Figure 2.3, it

appears to be very similar in construction to the high level SISO architecture defined in

Figure 2.1. The key difference between the two systems, other than the inclusion of

multiple transmit and receive antennas, is the receiver’s method used to estimate the

transmitted signal. In the SISO architecture, an equalizer is used in accordance with the

process as described in the SISO OFDM description. In the MIMO case, an architecture

developed by Bell Laboratories known as Vertical-Bell Laboratories-Layered-Space-

Time (V-Blast) [5] is utilized to estimate the transmitted signal. Before diving into the

details of V-Blast, the reason for considering MIMO systems must be introduced. The

12

obvious advantage to MIMO systems is the fact that system throughput increases as the

total number of transmitters and receivers increases, while occupying an amount of

bandwidth consistent with SISO OFDM systems. For example, in a simple two

transmitter, two receiver system, different data is transmitted by the first antenna and the

second antenna in the same time slot and at the same frequency. Thus, in this simple

example, the data rate is doubled with respect to a traditional SISO system. In general, it

has been proven that the channel capacity of MIMO systems is greater than that of SISO

systems [6]. The capacity of the SISO system in AWGN is defined by Equation 2.19 and

the capacity of a MIMO system is defined by Equation 2.20

𝐶𝑆𝐼𝑆𝑂 = log2 1 + 𝑆𝑁𝑅 (2.19)

𝐶𝑀𝐼𝑀𝑂 = 𝑙𝑜𝑔2 𝑑𝑒𝑡 𝑰 +𝑆𝑁𝑅

𝑁𝒉𝒉𝑇∗ (2.20)

Where:

𝐶𝑆𝐼𝑆𝑂 = SISO Capacity (bits/s/Hz)

𝐶𝑀𝐼𝑀𝑂 = MIMO Capacity (bits/s/Hz)

𝑆𝑁𝑅 = Signal-to-Noise Ratio (Linear)

𝑀 = Number of Receive Antennas

𝑁 = Number of Transmit Antennas

𝑰 = NxM Identity Matrix

𝒉 = NxM MIMO Fading Channel

In order to visualize the difference in capacity expressed by Equation 2.19 and Equation

2.20, Figure 2.4 has been constructed to show both the SISO and MIMO capacities for a

random complex fading channel.

13

Fig. 2.4. SISO Capacity vs. MIMO Capacity

The V-Blast MIMO architecture offers many benefits to which advantage can be

taken. The primary benefit employed by the MIMO architecture utilized in this analysis

is spatial multiplexing gain. Spatial multiplexing gain is achieved through utilization of a

rich scattering/fading environment that allows for each transmitter to utilize the same

carrier frequency and transmission power, or in the case of OFDM, the same carrier

frequencies [5]. In this design, maximization of throughput can be achieved if the

channel environment is dynamic enough to allow the receiver to discern between signals

received from each transmitter. V-Blast is a specific approach for MIMO systems that

aims to take advantage of spatial multiplexing gain and maximize throughput [5]. This is

-5 0 5 10 15 200

1

2

3

4

5

6

SNR (dB)

Capacity (

bits/s

/Hz)

SISO Capacity

MIMO Capacity

14

achieved via an algorithm that resides in the receiver and utilizes the signals received

from both antennas in order to determine an estimate for the transmitted signal.

2.4 V-Blast MIMO OFDM Description

To begin a detailed discussion, consider the MIMO OFDM multi-carrier

modulation system with 𝑚 transmit and 𝑛 receive antennas, as illustrated in Figure 2.5.

IFFT P/S

X1(0)

X1(1)

X1(N-1)

.

.

.

.

.

.

S/P FFT.

.

.

VBLAST

Symbol

Detection

.

.

.

.

.

.

Y1(0)

Y1(1)

Y1(N-1)

IFFT

Xm(0)

Xm(1)

Xm(N-1)

.

.

.

.

.

.

Yn(0)

Yn(1)

Yn(N-1)

Add

CP

h11 +

Remove

CP

P/S.

.

.

S/P FFT.

.

.

.

.

.

Add

CP

hnm +

Remove

CP

hn1 +

h1m +

w

w

Fig. 2.5. V-Blast MIMO OFDM Receiver/Transmitter Block Diagram

It can be observed that the system represented in Figure 2.5 appears to be more

complicated than the MIMO OFDM system illustrated in Figure 2.3. The added

complication is due to the fact that the system modeled in Figure 2.5 includes a reference

to the OFDM CP (see Chapter 2.2 for more information) as well as the MIMO channel

model. The complex information symbols 𝑿𝒊 associated with the 𝑖𝑡𝑕 transmitter are

shown in Equation 2.21.

15

𝑿𝒊 =

𝑋𝑖 0

𝑋𝑖 1 ⋮

𝑋𝑖(𝑁 − 1)

(2.21)

Each set of complex symbols 𝑿𝒊 is derived from symbol array 𝑿, which is defined as

follows.

𝑿 =

𝑿𝟏

𝑿𝟐

⋮𝑿𝒎

(2.22)

If the total number of transmitters is equal to three, then 𝑚 equals three and 𝑿 has a

vector length 3𝑁, where 𝑁 is the total number of carriers associated with any transmitter.

In order to define each 𝑿𝒊, 𝑿 is parsed into 𝑚 data vectors of equal length, such that

different sets of complex symbols can be transmitted in parallel. The values associated

with the complex symbols are derived from bit-to-symbol mapping techniques such as

QPSK or QAM. Similar to the SISO case, the IFFT blocks represented in Figure 2.5

provide the capability to transform the complex information symbols associated with a

specific transmitter, into a time domain representation via standard algorithm. Inclusion

of the IFFT algorithm ensures orthogonality between the carriers of a specific transmitter.

In this particular description, the IFFT length 𝑁 is equal to the number of carriers

associated with any 𝑿𝒊. In order to provide a mathematical representation of the IFFT,

the notation 𝑭𝑵−1 is introduced to represent the IDFT matrix of size 𝑁𝑥𝑁. As such, for 𝑚

transmit antennas, the following is declared where notation ⨂ is the Kronecker Product

and 𝑰𝒎 is an Identity matrix with a size of 𝑚𝑥𝑚 [7].

𝐱 = (𝑭𝑵−1⨂𝑰𝒎)𝑿 (2.23)

16

Each 𝒙𝒊 is the length 𝑁 time domain representation of 𝑿𝒊. Progressing through Figure

2.5, a cyclic prefix (CP) of length 𝐾 is applied to each 𝒙𝒊. In this particular analysis, it is

assumed that the channel impulse response duration associated with each permutation of

transmitter and receiver are the same. Taking this into account, the inclusion of the CP

results in an 𝒙𝒊 length equal to 𝑁 + 𝐾. The process of applying the CP to each vector 𝒙𝒊

is described as follows where 𝑇 is the transpose operation.

𝒙𝒄𝒑𝒊𝑇 =

𝑥𝑖 𝑁 − 𝐾 , 𝑥𝑖 𝑁 − 𝐾 + 1 , …𝑥𝑖 𝑁 − 1 , 𝑥𝑖 0 , 𝑥𝑖 1 , … , 𝑥𝑖 𝑁 − 1 (2.24)

The CP length, previously defined as 𝐾, is determined by the channel characterization

associated with every possible spatial combination of transmit and receive antennas. In

order to determine an optimum value of 𝐾, the MIMO wireless channel must be

estimated. The channel can be modeled as a matrix of coefficients in accordance with

every possible permutation of transmit and receive antennas. Equation 2.25 depicts the

MIMO channel generically for 𝑚 transmitters and 𝑛 receivers. Each specific channel

coefficient 𝑕𝑗𝑖 , where 𝑗 identifies the receiver and 𝑖 identifies the transmitter, is a complex

Gaussian random variable that provides the fading gain for every spatial path of

transmission.

𝒉 =

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

(2.25)

Similar to the SISO analysis, it is assumed that the MIMO channel can be characterized

by slow fading and thus the channel impulse response does not change within one OFDM

symbol. This analysis also does not consider the multi-path associated with each specific

17

transmitter and receiver combination and thus, CP is not actually required. In general

though, the same principles used to define the length of the CP for the SISO case also

apply to MIMO. Therefore, the CP length must exceed the duration of the channel

impulse response, with 𝐿 channel coefficients as seen by each combination of transmit

and receive antennas, defined in Equation 2.26.

𝐾 ≥ 𝐿 − 1 (2.26)

Once the CP is incorporated into 𝒙 to define 𝒙𝒄𝒑, the transmission of data into the MIMO

channel occurs. At each receiver in the MIMO system, it is assumed that ISI occurs for

the first 𝐾 received symbols (the CP symbols) and thus these symbols included in

received signal 𝒚𝒄𝒑 are discarded. The procedure for removing the CP from 𝒚𝒄𝒑 to

define signal 𝒚 is presented in Equation 2.27.

𝒚 =

𝒚𝒄𝒑𝟏 𝐾 − 1 : 𝑁 + 𝐾 − 1

⋮𝒚𝒄𝒑𝒎 𝐾 − 1 : (𝑁 + 𝐾 − 1)

(2.27)

With the elimination of the CP by the MIMO receiver, signal 𝒚 can be mathematically

represented as the linear convolution between the transmitted signal 𝒙 and associated

MIMO channel coefficient, plus channel noise 𝒘, as specified in Equation 2.28.

𝒚𝟏

𝑇

𝒚𝟐𝑇

⋮𝒚𝒏

𝑇

=

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

∗

𝒙𝟏

𝑇

𝒙𝟐𝑇

⋮𝒙𝒎

𝑇

+

𝒘𝟏

𝑇

𝒘𝟐𝑇

⋮𝒘𝒏

𝑇

(2.28)

Channel noise 𝒘 included in Equation 2.28 exists for each spatial path and is Additive

White Gaussian Noise (AWGN) with zero mean and variance 𝜎2 =𝑁0

2, where 𝑁0 is the

18

single-sided power spectral density [1]. The channel noise associated with the 𝑗𝑡𝑕 receive

antenna can be represented as follows.

𝒘𝒋 =

𝑤𝑗 0

𝑤𝑗 1

⋮𝑤𝑗 (𝑁 − 1)

(2.29)

At this point, the FFT is applied to the received signal 𝒚 to reverse the IFFT modulation

in the transmitter. Similar to the system’s implementation of the IFFT in the transmitter,

the FFT is also of length 𝑁. In order to provide a mathematical representation of the

FFT, the notation 𝑭𝑵 is introduced in Equation 2.30 in order to represent the Discrete

Fourier Transform (DFT) matrix of size 𝑁𝑥𝑁.

𝒀 = (𝑭𝑵⨂𝑰𝒏)𝒚 (2.30)

After computing the DFT to define 𝒀, an approach is subsequently determined to recover

an estimate of 𝑿 from 𝒀. Analyzing the assumptions made with respect to the MIMO

OFDM system, it can be concluded that the linear convolution between the MIMO

channel matrix 𝒉 and transmitted signal 𝒙 can be rewritten with multiplication as follows

due to the singular duration of channel impulse response.

𝒚𝟏

𝒚𝟐

⋮𝒚𝒏

=

𝑕11𝒙𝟏 + 𝑕12𝒙𝟐 + ⋯ + 𝑕1𝑚𝒙𝒎

𝑕21𝒙𝟏 + 𝑕22𝒙𝟐 + ⋯ + 𝑕2𝑚𝒙𝒎

⋮𝑕𝑛1𝒙𝟏 + 𝑕𝑛2𝒙𝟐 + ⋯ + 𝑕𝑛𝑚 𝒙𝒎

+

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(2.31)

With this arrangement, the relationship expressed in Equation 2.30 can be substituted into

Equation 2.31 as follows.

𝒀𝟏

𝒀𝟐

⋮𝒀𝒏

= (𝑭𝑵⨂𝑰𝒏)

𝑕11𝒙𝟏 + 𝑕12𝒙𝟐 + ⋯ + 𝑕1𝑚𝒙𝒎 + 𝒘𝟏

𝑕21𝒙𝟏 + 𝑕22𝒙𝟐 + ⋯ + 𝑕2𝑚𝒙𝒎 + 𝒘𝟐

⋮𝑕𝑛1𝒙𝟏 + 𝑕𝑛2𝒙𝟐 + ⋯ + 𝑕𝑛𝑚 𝒙𝒎 + 𝒘𝒏

(2.32)

19

Furthermore, each signal 𝒙𝒊 is equivalent to the IDFT of corresponding vector 𝑿𝒊 and as

such, can be utilized as indicated in Equation 2.33.

𝒀𝟏

𝒀𝟐

⋮𝒀𝒏

=

(𝑭𝑵⨂𝑰𝒏)

𝑕11(𝑭𝑵

−1𝑿𝟏) + 𝑕12(𝑭𝑵−1𝑿𝟐) + ⋯ + 𝑕1𝑚(𝑭𝑵

−1𝑿𝒎) + 𝒘𝟏

𝑕21(𝑭𝑵−1𝑿𝟏) + 𝑕22(𝑭𝑵

−1𝑿𝟐) + ⋯ + 𝑕2𝑚 (𝑭𝑵−1𝑿𝒎) + 𝒘𝟐

⋮𝑕𝑛1(𝑭𝑵

−1𝑿𝟏) + 𝑕𝑛2(𝑭𝑵−1𝑿𝟐) + ⋯ + 𝑕𝑛𝑚 (𝑭𝑵

−1𝑿𝒎) + 𝒘𝒏

(2.33)

Using the properties of the matrix formed by the Kronecker Product, the DFT matrix 𝑭𝑵

can be transitioned into Equation 2.33 as follows.

𝒀𝟏

𝒀𝟐

⋮𝒀𝒏

=

𝑕11(𝑭𝑵𝑭𝑵

−1𝑿𝟏) + 𝑕12(𝑭𝑵𝑭𝑵−1𝑿𝟐) + ⋯ + 𝑕1𝑚 (𝑭𝑵𝑭𝑵

−1𝑿𝒎) + (𝑭𝑵𝒘𝟏)

𝑕21(𝑭𝑵𝑭𝑵−1𝑿𝟏) + 𝑕22(𝑭𝑵𝑭𝑵

−1𝑿𝟐) + ⋯ + 𝑕2𝑚 (𝑭𝑵𝑭𝑵−1𝑿𝒎) + (𝑭𝑵𝒘𝟐)

⋮𝑕𝑛1(𝑭𝑵𝑭𝑵

−1𝑿𝟏) + 𝑕𝑛2(𝑭𝑵𝑭𝑵−1𝑿𝟐) + ⋯ + 𝑕𝑛𝑚 (𝑭𝑵𝑭𝑵

−1𝑿𝒎) + (𝑭𝑵𝒘𝒏)

(2.34)

Additional rearrangement can be made given the fact that a matrix multiplied by its

inverse results in an Identity matrix. Utilizing said property, Equation 2.34 can be

reduced to the following.

𝒀𝟏

𝒀𝟐

⋮𝒀𝒏

=

𝑕11𝑿𝟏 + 𝑕12𝑿𝟐 + ⋯ + 𝑕1𝑚𝑿𝒎

𝑕21𝑿𝟏 + 𝑕22𝑿𝟐 + ⋯ + 𝑕2𝑚𝑿𝒎

⋮𝑕𝑛1𝑿𝟏 + 𝑕𝑛2𝑿𝟐 + ⋯ + 𝑕𝑛𝑚𝑿𝒎

+ (𝑭𝑵⨂𝑰𝒏)

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(2.35)

At this point, symbol 𝑾 is defined as follows to represent the frequency representation of

AWGN.

20

𝑾 =

𝑾𝟏

𝑾𝟐

⋮𝑾𝒏

= (𝑭𝑵⨂𝑰𝒏)

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(2.36)

For the purposes of simplicity with the explanation going forward, vectors 𝒀, 𝑿 and 𝑾

are redefined as follows.

𝒀 =

𝒀𝟏

𝑇

𝒀𝟐𝑇

⋮𝒀𝒏

𝑇

, 𝑿 =

𝑿𝟏

𝑇

𝑿2𝑇

⋮𝑿𝒎

𝑇

, 𝑾 =

𝑾𝟏

𝑇

𝑾𝟐𝑇

⋮𝑾𝒏

𝑇

(2.37)

With the definition of Equation 2.37, the MIMO OFDM communication system can be

represented in a format easily extended for subsequent processing as shown in Equation

2.38.

𝒀 =

𝒀𝟏

𝑇

𝒀𝟐𝑇

⋮𝒀𝒏

𝑇

=

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

𝑿𝟏

𝑇

𝑿𝟐𝑇

⋮𝑿𝒎

𝑇

+

𝑾𝟏

𝑇

𝑾𝟐𝑇

⋮𝑾𝒏

𝑇

= 𝒉𝑿 + 𝑾 (2.38)

With Equation 2.38, the communications model has be simplified to the point where an

estimate of 𝑿 can be determined using a standard V-Blast approach. MIMO V-Blast

offers a few different alternatives that can be employed in order to determine a viable

estimate of 𝑿. The focus of this research has been specific to the V-Blast algorithms of

Successive Interference Cancellation (SIC) with Optimal Ordering using both Zero-

Forcing (ZF) and Minimum Mean Square Error (MMSE) equalization. The following

sections provide a technical overview of the equalization techniques of ZF and MMSE as

well as a description of optimal ordered SIC. Once an estimate of 𝑿 is determined using

one of the aforementioned V-Blast techniques, the BER is computed in order to measure

the communication system performance.

21

2.4.1 V-Blast ZF equalization

V-Blast ZF equalization provides a simple approach to determine a realistic

estimate of transmitted signal 𝑿. In order to successfully solve for an estimate of 𝑿,

spatial filtering matrix 𝑾𝒁𝑭 is computed as follows through utilization of the MIMO

channel model where notation 𝑇 ∗ is the conjugate transpose [8].

𝑾𝒁𝑭 = (𝒉𝑇∗𝒉)−1 𝒉𝑇∗ (2.39)

After matrix 𝑾𝒁𝑭 is determined, it is applied to 𝒀 to define the following.

𝑿 = 𝑾𝒁𝑭𝒀 = (𝒉𝑇∗𝒉)−1 𝒉𝑇∗ 𝒉𝑿 + 𝑾 (2.40)

Looking in detail at the relationship contained in Equation 2.40, it can noted that the

estimate of 𝑿, defined as 𝑿 , contains an additive ratio of noise applied to equalizing

matrix 𝑾𝒁𝑭 as clarified by Equation 2.41.

𝑿 = 𝑿 + (𝒉𝑇∗𝒉)−1 𝒉𝑇∗𝑾 (2.41)

In order to reduce the impact of the additive ratio of noise to equalization and improve

the estimate of 𝑿, optimal ordered successive interference cancellation is employed.

2.4.2 V-Blast MMSE equalization

Similar to the algorithm based on ZF equalization as described in the previous

section, MMSE equalization is applied to the received signal 𝒀 via a spatial filtering

matrix defined as 𝑾𝑴𝑴𝑺𝑬. MMSE equalization provides an approach that is more

accurate than ZF and thus allows for computation of a more realistic estimate of

transmitted signal 𝑿. In order to successfully solve for an estimate of 𝑿, spatial filtering

22

matrix 𝑾𝑴𝑴𝑺𝑬 is computed as follows through utilization of the MIMO channel model

where 𝑁0 is the single-sided noise power spectral density and 𝑰 is the identity matrix [8].

𝑾𝑴𝑴𝑺𝑬 = (𝒉𝑇∗𝒉+𝑁0𝑰)−1 𝒉𝑇∗ (2.42)

When comparing the MMSE equalizing matrix to ZF equalization, it is important to note

that 𝑾𝑴𝑴𝑺𝑬 contains an additive component dependent on noise. The goal of MMSE is

to develop matrix 𝑾𝑴𝑴𝑺𝑬 to minimize the error between transmitted signal 𝑿 and

received signal 𝒀 as follows.

𝐸𝑟𝑟𝑜𝑟 = 𝑾𝑴𝑴𝑺𝑬𝒀 − 𝑿 𝑾𝑴𝑴𝑺𝑬𝒀 − 𝑿 𝑇∗ (2.43)

After matrix 𝑾𝑴𝑴𝑺𝑬 is computed, it is applied to 𝒀 as described in Equation 2.44.

𝑿 = 𝑾𝑴𝑴𝑺𝑬𝒀 = (𝒉𝑇∗𝒉+𝑁0𝑰)−1 𝒉𝑇∗ 𝒉𝑿 + 𝑾 (2.44)

The result of MMSE is declared as in Equation 2.45 where the cumulative error

associated with 𝑿 due to the channel and AWGN is minimized.

𝑿 = (𝒉𝑇∗𝒉+𝑁0𝑰)−1 𝒉𝑇∗𝒉𝑿 + (𝒉𝑇∗𝒉+𝑁0𝑰)−1 𝒉𝑇∗𝑾 (2.45)

Similar to ZF equalization the remaining error associated with 𝑿 can be further reduced

using optimal ordered successive interference cancellation.

2.4.3 SIC with optimal ordering

The first step in the implementation of optimal ordered successive interference

cancellation is to determine the transmitted array 𝑿𝒊 that most likely was received with

the minimum collective power across all receiver antennas [5]. This is determined by

assessing the magnitude of each MIMO channel coefficient with respect to a specific

transmitted signal 𝑿𝒊 as shown in Equation 2.46.

23

𝑷𝒙 = 𝑃𝑥1

⋮𝑃𝑥𝑚

= 𝑕11

2 + 𝑕21 2 + ⋯ + 𝑕𝑛1

2

⋮ 𝑕1𝑚 2 + 𝑕2𝑚 2 + ⋯ + 𝑕𝑛𝑚 2

(2.46)

After the computation of 𝑷𝒙, the 𝑿𝒊 associated with the 𝑃𝑥𝑖 that is the minimum of vector

𝑷𝒙, will be estimated first using traditional SIC. The SIC algorithm with optimal

ordering ensures that the first estimate of 𝑿 will have a lower probability of error than

any other symbol estimate. As the error probability associated with a symbol estimate

decreases, the likelihood of making incorrect decisions in the receiver decreases. For the

purposes of this description, the estimate of 𝑿𝒊 with associated minimum 𝑃𝑥𝑖 is declared

as 𝑿 𝒊_𝒎𝒊𝒏. The process for estimating 𝑿 𝒊_𝒎𝒊𝒏 with SIC requires the subtraction of all

other values of 𝑿 , multiplied by the appropriate channel coefficient as indicated by

Equation 2.47.

𝑹 =

𝑹𝟏

𝑹𝟐

⋮𝑹𝒏

=

𝒀𝟏

𝑇

𝒀𝟐𝑇

⋮𝒀𝒏

𝑇

+

−𝑕11𝑿 𝟏

𝑇 − ⋯− 𝑕1𝑚𝑿 𝒎𝑇 + 𝑕1𝑖𝑿 𝒊_𝒎𝒊𝒏

𝑇

−𝑕21𝑿 𝟏𝑇 − ⋯− 𝑕2𝑚𝑿 𝒎

𝑇 + 𝑕2𝑖𝑿 𝒊_𝒎𝒊𝒏𝑇

⋮−𝑕𝑛1𝑿 𝟏

𝑇 − ⋯− 𝑕𝑛𝑚𝑿 𝒎𝑇 + 𝑕𝑛𝑖𝑿 𝒊_𝒎𝒊𝒏

𝑇

(2.47)

It is important to note that Equation 2.47 includes the additive term 𝑿 𝒊_𝒎𝒊𝒏 multiplied by

its associated channel coefficient in order to clearly show that it is not subtracted from 𝒀

like all other vectors of 𝑿 . Substituting the definition of 𝒀 into Equation 2.47 results in

the following.

24

𝑹𝟏

𝑹𝟐

⋮𝑹𝒏

=

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

𝑿𝟏

𝑇

𝑿𝟐𝑇

⋮𝑿𝒏

𝑇

+ 𝑾 +

−𝑕11𝑿 𝟏


𝑇

−𝑕21𝑿 𝟏𝑇 − ⋯− 𝑕2𝑚𝑿 𝒎


⋮−𝑕𝑛1𝑿 𝟏


𝑇

(2.48)

It can be assumed that the cumulative error included in all values of 𝑿 approximately

accounts for the additive AWGN present in received signal 𝒀 as shown in Equation 2.49

and thus, 𝑿 𝒊_𝒎𝒊𝒏 can be determined in an iterative fashion using the format expressed in

Equation 2.50.

𝑕11𝑿𝟏

𝑇 + 𝑕12𝑿𝟐𝑇 + ⋯ + 𝑕1𝑚𝑿𝒎

𝑇

𝑕21𝑿𝟏𝑇 + 𝑕22𝑿𝟐

𝑇 + ⋯ + 𝑕2𝑚𝑿𝒎𝑇

⋮𝑕𝑛1𝑿𝟏

𝑇 + 𝑕𝑛2𝑿𝟐𝑇 + ⋯ + 𝑕𝑛𝑚𝑿𝒎

𝑇

+ 𝑊 +

−𝑕11𝑿 𝟏


𝑇

−𝑕21𝑿 𝟏𝑇 − ⋯− 𝑕2𝑚𝑿 𝒎


⋮−𝑕𝑛1𝑿 𝟏


𝑇

≅

𝑕1𝑖𝑿 𝒊_𝒎𝒊𝒏

𝑇

𝑕2𝑖𝑿 𝒊_𝒎𝒊𝒏𝑇

⋮𝑕𝑛𝑖𝑿 𝒊_𝒎𝒊𝒏

𝑇

(2.49)

𝑅1(𝑘)

⋮𝑅𝑛(𝑘)

= 𝑕1𝑖

⋮𝑕𝑛𝑖

𝑋 𝑖_𝑚𝑖𝑛 𝑘 , 𝑤𝑕𝑒𝑟𝑒 𝑘 = 0. . 𝑁 − 1 (2.50)

Specifically, complex symbols 𝑿 𝒊_𝒎𝒊𝒏 can be estimated by Maximal Ratio Combining

(MRC) through rearrangement of Equation 2.50 as indicated by Equation 2.51.

𝑋 𝑖_𝑚𝑖𝑛 (𝑘) =

𝑕1𝑖⋮

𝑕𝑛𝑖

𝑇∗

𝑅1(𝑘)

⋮𝑅𝑛 (𝑘)

𝑕1𝑖⋮

𝑕𝑛𝑖

𝑇∗

𝑕1𝑖⋮

𝑕𝑛𝑖

(2.51)

25

Once all values of array 𝑿 𝒊_𝒎𝒊𝒏 are computed, the remaining estimates of 𝑿 can be

determined by repeating the process defined in this section for each transmitted 𝑿𝒊. The

process re-initiates after each estimate by determining the next value of 𝑿 to be computed

based on the smallest value of 𝑷𝒙 for which an associated estimate of 𝑿 has not already

been determined. For each subsequent estimate of 𝑿, all 𝑿𝒊 that have already been

computed via SIC are utilized in place of the original estimates determined with ZF and

MMSE equalization.

26

3. SIMPLIFIED SYSTEM

3.1 Simple System Description

With the conventional system clearly defined, this chapter introduces the

approach necessary to describe the simplified system. The use of the word “simplified”

in this context directly pertains to a reduction in computational complexity associated

with multi-carrier systems, such as OFDM. The reduction of computational complexity

is derived from the application of a new approach in performing the Fourier Transform

and Inverse Fourier Transform. As indicated in the previous chapter, the OFDM

implementation of each transmitter and receiver requires the use of the transform in order

to ensure carrier orthogonality and to simplify equalization. Extending the simplification

to MIMO systems that utilize OFDM modulation, where each receiver utilizes an FFT

and each transmitter utilizes an IFFT, the simplified approach can provide significant

savings in complexity. The approach utilized to execute both the Fourier Transform and

Inverse Fourier Transform introduces the capability to do so by performing all

multiplications with values of negative one, zero, one and powers of two. Such

multiplications are very simple to implement and are considered trivial. In doing so, the

simplification results in multiplier-less versions of Fourier Transform and Inverse Fourier

Transform. The multiplier-less transforms are derived from the process of intelligently

quantizing functions sin(𝑥) and cos(𝑥) included in the general equation necessary to

27

describe the DFT. Proper utilization of the rounded functions allow for simpler

algorithms in terms of multiplicative complexity.

3.2 Simple Discrete Fourier Transform Matrix

The general equation for determining the DFT of array 𝒙 is defined by Equation

3.1 [9].

𝑋𝑘 = 𝑥𝑛𝑒−2𝜋𝑗𝑘𝑛

𝑁𝑁−1𝑛=0 𝑤𝑕𝑒𝑟𝑒 𝑘 = 0, … , 𝑁 − 1 (3.1)

The process for determining the DFT can also be represented in a matrix format in

accordance with Equation 3.2 and Equation 3.3, where 𝑭𝑵 represents the conventional

DFT matrix and 𝜔𝑁 is commonly referred to as the twiddle factor.

𝑭𝑵 = (𝜔𝑁𝑘𝑛 )𝑘 ,𝑛=0,…,𝑁−1 (3.2)

𝜔𝑁 = 𝑒−𝑗2𝜋

𝑁 (3.3)

With the definition of matrix 𝑭𝑵, it can be used to determine the DFT of vector 𝒙 as

indicated by Equation 3.4.

𝑿 = 𝑭𝑵𝒙 (3.4)

In order to derive the simplified version of matrix 𝑭𝑵, first recall Euler’s Identity as


𝑒𝑗𝑥 = cos 𝑥 + 𝑗 sin(𝑥) (3.5)

Substituting Euler’s Identity into Equation 3.3, the twiddle factor can be represented as


𝜔𝑁 = cos 2𝜋

𝑁 − 𝑗 sin

2𝜋

𝑁 (3.6)

28

Furthermore, the relationship defined in Equation 3.6 can be substituted into Equation 3.2

as shown by Equation 3.7, to represent the DFT.

𝑭𝑵 = cos 2𝜋𝑘𝑛

𝑁 − 𝑗 sin

2𝜋𝑘𝑛

𝑁

𝑘 ,𝑛=0,…,𝑁−1 (3.7)

With the definition of Equation 3.7, the DFT matrix 𝑭𝑵 is in the proper format to apply

the simplification.

In order to represent the rounded sin(𝑥) and cos(𝑥) functions, the following

syntax is introduced, where 𝑟𝑜𝑢𝑛𝑑() is the round-off operation and 𝑘 indicates the level

of quantization [10].

𝑟𝑐𝑜𝑠𝑘 𝑥 =𝑟𝑜𝑢𝑛𝑑 (𝑘 cos 𝑥 )

𝑘 (3.8)

𝑟𝑠𝑖𝑛𝑘 𝑥 =𝑟𝑜𝑢𝑛𝑑 (𝑘 sin 𝑥 )

𝑘 (3.9)

As the value of 𝑘 is increased, the response of functions 𝑟𝑠𝑖𝑛𝑘 𝑥 and 𝑟𝑐𝑜𝑠𝑘 𝑥 approach

the behavior of the conventional sin(𝑥) and cos(𝑥) functions [10]. Figure 3.1 displays

the response of 𝑟𝑠𝑖𝑛𝑘 𝑥 with respect to the conventional sin(𝑥), where k = 2.

29

Fig. 3.1. Response of Five Level Rounded Sine (k=2)

Increasing the value of 𝑘 to eight and comparing the response of 𝑟𝑠𝑖𝑛𝑘 𝑥 with sin(𝑥) as

shown in Figure 3.2, it is observed that the response of 𝑟𝑠𝑖𝑛𝑘 𝑥 more closely

approximates sin(𝑥).

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Am

plit

ude

Phase (radians)

Conventional Sine

Quantized Sine

30

Fig. 3.2. Response of Seventeen Level Rounded Sine (k=8)

In general, the number of quantization steps represented in the result of computing

functions 𝑟𝑠𝑖𝑛𝑘 𝑥 and 𝑟𝑐𝑜𝑠𝑘 𝑥 can be described by the relationship defined in

Equation 3.10.

𝑄𝑠𝑡𝑒𝑝𝑠 = 2𝑘 + 1 (3.10)

With an understanding of the performance associated with the rounded functions,

it is time to apply the quantization directly to the DFT matrix. To do so, the 𝑟𝑠𝑖𝑛𝑘 𝑥

and 𝑟𝑐𝑜𝑠𝑘 𝑥 functions are substituted into Equation 3.7 as shown by Equation 3.11

where notation 𝑭 𝑵 represents the rounded DFT [10].

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Am

plit

ude

Phase (radians)

Conventional Sine

Quantized Sine

31

𝑭 𝑵 = rcosk 2𝜋𝑘𝑛

𝑁 − 𝑗 rsink

2𝜋𝑘𝑛

𝑁

𝑘 ,𝑛=0,…,𝑁−1 (3.11)

In order to achieve the system level simplification expected of this research, the values of

𝑘 that can be selected must be done intelligently. In binary digital arithmetic,

multiplications with values of negative one, zero, one and powers of two are very simply

computed, implemented and considered trivial. As such, the simplification proposed by

Equation 3.11 can be optimized by selecting values of 𝑘 that will result in a DFT matrix

that consists entirely of values that provide trivial multiplications and thus permitting the

matrix to be considered multiplier-less. More specifically, values of 𝑘 are desired such

that the responses of 𝑟𝑠𝑖𝑛𝑘 𝑥 and 𝑟𝑐𝑜𝑠𝑘 𝑥 include as many powers of two as possible.

This is achieved by utilizing values of 𝑘 that are in fact a power of two. Equation 3.12

provides the guideline for selecting 𝑘.

𝑘 = 2𝑛 𝑤𝑕𝑒𝑟𝑒 𝑛 = 1,2, … ,∞ (3.12)

Simply utilizing values of 𝑘 in accordance with Equation 3.12 will not completely result

in a matrix that consists of values that support trivial multiplications. For example, when

𝑘 = 4, the DFT matrix will contain values of 0.75, which of course are not a power of

two. There is a method though that will permit DFT matrices of all possible values of 𝑘,

to be considered multiplier-less. Reconsidering the rounded DFT matrix with 𝑘 = 4,

even though a value of 0.75 is not a power of two, it can be obtained via the addition or

subtraction two values that are in fact powers of two such as one minus 0.25 or 0.25 plus

0.50. With this observation, it can be stated that a multiplier-less DFT matrix for all

values of 𝑘 can be developed by increasing the total number of additions. In the

example where 𝑘 = 4, every instance of 0.75 in the DFT matrix will result in two

32

additions as opposed to one non-trivial multiplication. A few subsequent additions can

be tolerated as in general, binary multiplication is more difficult to implement than binary

addition. Consider the terminology (𝑛) 𝐴𝑇2𝛼 𝑀 and (𝑛) 𝐴𝑇2𝛼 𝐴 to represent the

minimum area-time digital hardware complexity for 𝑛-bit multiplication and addition

respectively with the following notation [11].

𝐴 = 𝐶𝑕𝑖𝑝 𝐴𝑟𝑒𝑎

𝑇 = 𝑃𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒 𝑇𝑖𝑚𝑒

𝛼 ∈ 0, 1

When comparing (𝑛) 𝐴𝑇2𝛼 𝑀 with (𝑛) 𝐴𝑇2𝛼 𝐴, as shown in Equation 3.13, the result is

a multiplicative complexity that is on the order of 𝑛 greater than additive complexity.

(𝑛) 𝐴𝑇2𝛼 𝑀

(𝑛) 𝐴𝑇2𝛼 𝐴= Ω 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼 (3.13)

3.3 Simple Inverse Discrete Fourier Transform Matrix

Because the transmitter in an OFDM system utilizes the IDFT, simplification of

the IDFT matrix is also be considered. Utilizing an approach similar to the determination

of the rounded DFT, consider the general equation for the IDFT of an array 𝑿 as defined

by Equation 3.14.

𝑥𝑛 =1

𝑁 𝑋𝑘𝑒

2𝜋𝑖𝑘𝑛

𝑁𝑁−1𝑘=0 𝑤𝑕𝑒𝑟𝑒 𝑛 = 0, … , 𝑁 − 1 (3.14)

As in the case of the DFT, the process for determining the IDFT can be represented in a

matrix format in accordance with Equations 3.3.2 and 3.3.3 where 𝑭𝑵−1 represents the

conventional IDFT matrix and 𝜔𝑁 is the twiddle factor.

33

𝑭𝑵−𝟏 =

1

𝑁(𝜔𝑁

−𝑘𝑛 )𝑘 ,𝑛=0,…,𝑁−1 (3.15)

𝜔𝑁 = 𝑒−𝑗2𝜋

𝑁 (3.16)

Similar to the derivation for the rounded DFT, the rounded IDFT matrix, with notation

𝑭 𝑵−1, can be determined through utilization of the 𝑟𝑠𝑖𝑛𝑘 𝑥 and 𝑟𝑐𝑜𝑠𝑘 𝑥 functions to

define the matrix as in Equation 3.17.

𝑭 𝑵−1 =

1

𝑁 rcosk

2𝜋𝑘𝑛

𝑁 + 𝑗 rsink

2𝜋𝑘𝑛

𝑁

𝑘 ,𝑛=0,…,𝑁−1 (3.17)

Another method that can be utilized to determine the rounded DFT is to simply

compute the inverse of the rounded DFT matrix. Given the fact that a true inverse of the

rounded DFT cannot be easily computed for all values of 𝑁, another approach must be

utilized in order to determine the rounded IDFT matrix. To start, consider the fact that

any square matrix 𝑨 when applied to its inverse will result in an Identity matrix as shown

in Equation 3.18.

𝑨 ∙ 𝑨−1 = 𝑰 (3.18)

In order to determine the rounded IDFT, a matrix can be computed such that when

applied to the rounded DFT, the result is an approximate identity function as shown in

Equation 3.19.

𝑭 𝑵 ∙ 𝑭 𝑵−1 ≅ 𝑰 (3.19)

The approximate rounded IDFT matrix can be computed by taking the conjugate

transpose of the rounded DFT matrix as indicated by Equation 3.20.

𝑭 𝑵−1 =

1

𝑁𝑭 𝑵

𝑇∗ (3.20)

As such, an approximate, multiplier-less inverse matrix, can be determined directly from

the simplified implementation of the DFT [10]. Since the inverse is an approximation,

34

use of the IDFT computed in this manner will result in additional error in system

performance above and beyond the error injected due to quantization.

3.4 Simple Fast Fourier Transform Algorithm

Current system implementations do not simply perform matrix multiplication as

described in the previous two sections in order to utilize the Fourier Transform. As the

demand for improved performance increases, efficient algorithms have been introduced

to allow for fast and simple transformation. One such implementation of DFT is the

Radix-4 Fast Fourier Transform. The Radix-4 is derived by breaking up the original DFT

equation into four separate summations by providing 𝑁

4 consecutive samples of 𝒙 in each

sum as dictated by Decimation In Frequency (DIF) [9]. With some simplification, the

four summations can be recombined into the construct of a single summation as shown in

Equation 3.21, where 𝜔𝑁 represents the twiddle factor.

𝑋 𝑘 =

𝑥 𝑛 + (−𝑗)𝑘𝑥 𝑛 +𝑁

4 + (−1)𝑘𝑥 𝑛 +

𝑁

2 + (𝑗)𝑘𝑥 𝑛 +

3𝑁

4

𝑁/4 −1𝑛=0 𝜔𝑁

𝑛𝑘 (3.21)

In its current form, Equation 3.21 cannot be used to determine an FFT as the array length

is not consistently defined to be 𝑁/4 due to the definition of the twiddle factor that

depends on a length of 𝑁. In order to rearrange Equation 3.21 into an FFT of length 𝑁/

4, the sequence 𝑋 𝑘 is again divided into four separate summations for the cases of

𝑘 = 4𝑟, 𝑘 = 4𝑟 + 1, 𝑘 = 4𝑟 + 2 and 𝑘 = 4𝑟 + 3. With the property 𝜔𝑁4𝑛𝑘 = 𝜔𝑁/4

𝑛𝑘 , the

four sequences that comprise the Radix-4 FFT can defined by Equation 3.22, Equation

3.23, Equation 3.24 and Equation 3.25 [9].

35

𝑋 4𝑟 =

𝑥 𝑛 + 𝑥 𝑛 +𝑁

4 + 𝑥 𝑛 +

𝑁

2 + 𝑥 𝑛 +

3𝑁

4 𝜔𝑁

0 𝑁/4 −1𝑛=0 𝜔𝑁/4

𝑛𝑟 (3.22)

𝑋 4𝑟 + 1 =

𝑥 𝑛 − 𝑗𝑥 𝑛 +𝑁

4 − 𝑥 𝑛 +

𝑁

2 + 𝑗𝑥 𝑛 +

3𝑁

4 𝜔𝑁

𝑛 𝑁/4 −1𝑛=0 𝜔𝑁/4

𝑛𝑟 (3.23)

𝑋 4𝑟 + 2 =

𝑥 𝑛 − 𝑥 𝑛 +𝑁

4 + 𝑥 𝑛 +

𝑁

2 − 𝑥 𝑛 +

3𝑁

4 𝜔𝑁

2𝑛 𝑁/4 −1𝑛=0 𝜔𝑁/4

𝑛𝑟 (3.24)

𝑋 4𝑟 + 3 =

𝑥 𝑛 + 𝑗𝑥 𝑛 +𝑁

4 − 𝑥 𝑛 +

𝑁

2 − 𝑗𝑥 𝑛 +

3𝑁

4 𝜔𝑁

3𝑛 𝑁/4 −1𝑛=0 𝜔𝑁/4

𝑛𝑟 (3.25)

Each of the four equations listed above, collectively represent the Radix-4 FFT and can

be described as four length 𝑁/4 FFTs. Another method commonly used to visualize the

implementation of an FFT algorithm is a butterfly chart. A butterfly chart attempts to

visualize the processing that occurs in order to compute an FFT. As an example,

consider a length four Radix-4 FFT as shown in Figure 3.3.

36

j

j

-1

-1

-1

j

-1

j

x(n)

x(n + N/4)

x(n + N/2)

x(n + 3N/4)

X(k)

X(k + N/4)

X(k + N/2)

X(k + 3N/4)

+

+

+

+

ω0Nω

nkN/4

ωnkN/4ωn

N

ωnkN/4ω2n

N

ωnkN/4ω3n

N

Fig. 3.3. Four Point Radix-4 FFT Butterfly Diagram

Figure 3.3 clearly shows the dependency between each value of 𝑥 𝑛 on one another to

determine the FFT array 𝑿.

Utilizing concepts developed to simplify the DFT matrix, the same approach can

be applied to the implementation of the Radix-4 algorithm. Reviewing each of the four

equations that comprise the Radix-4 FFT, it can be observed that the only multiplications

are with respect to the twiddle factors associated with each 𝑁/4 FFT. Focusing

specifically on Equation 3.22, the collective twiddle factor is represented as in Equation

3.26.

𝑡𝑓1 = 𝜔𝑁0 𝜔𝑁/4

𝑛𝑟 (3.26)

To start, 𝑡𝑓1 can be further reduced as shown in Equation 3.27 as 𝜔𝑁0 = 1.

𝑡𝑓1 = 𝜔𝑁/4𝑛𝑟 (3.27)

37

Using the properties defined to create the rounded DFT matrix, a quantized version of 𝑡𝑓1

can be defined as follows.

𝑡𝑓 1 = rcosk 8𝜋𝑟𝑛

𝑁 − 𝑗 rsink

8𝜋𝑟𝑛

𝑁 (3.28)

With the definition of 𝑡𝑓 1, it can be applied to Equation 3.22 to result in the multiplier-

less sequence presented in Equation 3.29.

𝑋 4𝑟 = 𝑥 𝑛 + 𝑥 𝑛 +𝑁

4 + 𝑥 𝑛 +

𝑁

2 + 𝑥 𝑛 +

3𝑁

4

𝑁/4 −1𝑛=0 𝑡𝑓 1 (3.29)

The remaining three equations that comprise the Radix-4 are not as simply quantized.

Looking in detail with respect to Equation 3.23, the associated twiddle factor is defined

as in Equation 3.30.

𝑡𝑓2 = 𝜔𝑁𝑛𝜔𝑁/4

𝑛𝑟 (3.30)

Furthermore, twiddle factor 𝑡𝑓2 can be expanded as indicated in Equation 3.31.

𝑡𝑓2 = 𝜔𝑁𝑛𝜔𝑁/4

𝑛𝑟 = 𝑒−𝑗2𝑛𝜋

𝑁 𝑒−𝑗8𝑛𝑟𝜋

𝑁 = 𝑒−𝑗2𝑛𝜋

𝑁+

−𝑗8𝑛𝑟𝜋

𝑁 = 𝑒 −𝑗2𝑛𝜋

𝑁 1+4𝑟

(3.31)

Using the properties defined to create the rounded DFT matrix, a quantized version of 𝑡𝑓2

can be constructed as follows.

𝑡𝑓 2 = rcosk 2𝑛𝜋

𝑁 1 + 4𝑟 − 𝑗 rsink

2𝑛𝜋

𝑁 1 + 4𝑟 (3.32)

The definition of 𝑡𝑓 2 can be applied to Equation 3.23 resulting in the multiplier-less

sequence presented in Equation 3.33.

𝑋 4𝑟 + 1 = 𝑥 𝑛 − 𝑗𝑥 𝑛 +𝑁

4 − 𝑥 𝑛 +

𝑁

2 + 𝑗𝑥 𝑛 +

3𝑁

4

𝑁/4 −1𝑛=0 𝑡𝑓 2 (3.33)

Progressing with the same procedure, twiddle factors 𝑡𝑓 3 and 𝑡𝑓 4 can be represented as

shown in Equation 3.34 and Equation 3.35 and are used to define Equation 3.36 and

Equation 3.37 to fully represent the rounded FFT.

38



4𝑛𝜋

𝑁 1 + 2𝑟 (3.34)



2𝑛𝜋

𝑁 3 + 4𝑟 (3.35)

𝑋 4𝑟 + 2 = 𝑥 𝑛 − 𝑥 𝑛 +𝑁

4 + 𝑥 𝑛 +

𝑁

2 − 𝑥 𝑛 +

3𝑁

4

𝑁/4 −1𝑛=0 𝑡𝑓 3 (3.36)

𝑋 4𝑟 + 3 = 𝑥 𝑛 + 𝑗𝑥 𝑛 +𝑁

4 − 𝑥 𝑛 +

𝑁

2 − 𝑗𝑥 𝑛 +

3𝑁

4

𝑁/4 −1𝑛=0 𝑡𝑓 4 (3.37)

To conclude the definition of the rounded FFT, the butterfly diagram of Figure 3.3 has

been updated as shown in Figure 3.4, to clearly show the quantized twiddle factors.

j

j

-1

-1

-1

j

-1

j

x(n)

x(n + N/4)

x(n + N/2)

x(n + 3N/4)

X(k)

X(k + N/4)

X(k + N/2)

X(k + 3N/4)

+

+

+

+

Fig. 3.4. Four Point Rounded Radix-4 FFT Butterfly Diagram

3.5 Simple Inverse Fast Fourier Transform Algorithm

Similar to DFT, a “fast” version of the IDFT, referred to as the IFFT, can be

developed. In this description, the Radix-4 concept also provides the framework for

derivation of the rounded IFFT. The original definition of IDFT is broken into four

39

separate summations to have 𝑁/4 consecutive samples of 𝑿. With some simplification,

the four summations can be recombined into the construct of a single summation as

shown in Equation 3.38 where 𝜔𝑁 represents the twiddle factor.

𝑥 𝑛 =

𝑋 𝑘 + (𝑗)𝑛𝑋 𝑘 +𝑁

4 + (−1)𝑛𝑥 𝑘 +

𝑁

2 + (−𝑗)𝑛𝑥 𝑘 +

3𝑁

4

𝑁/4 −1𝑘=0 𝜔𝑁

−𝑛𝑘 (3.38)

In its current form, Equation 3.38 cannot be used to determine an IFFT as the array length

is not consistently defined to be 𝑁/4 as the twiddle factor depends on a length of 𝑁. In

order to rearrange Equation 3.38 into an IFFT of length 𝑁/4, the sequence x 𝑛 is

divided into four separate summations for the cases of 𝑛 = 4𝑟, 𝑛 = 4𝑟 + 1, 𝑛 = 4𝑟 + 2

and 𝑛 = 4𝑟 + 3. Noting the property 𝜔𝑁−4𝑛𝑘 = 𝜔𝑁/4

−𝑛𝑘 , the four sequences that comprise

the Radix-4 IFFT are defined by Equation 3.39, Equation 3.40, Equation 3.41 and

Equation 3.42.

𝑥 4𝑟 =

𝑋 𝑘 + 𝑋 𝑘 +𝑁

4 + 𝑋 𝑘 +

𝑁

2 + 𝑋 𝑘 +

3𝑁

4 𝜔𝑁

−0 𝑁/4 −1𝑘=0 𝜔𝑁/4

−𝑟𝑘 (3.39)

𝑥 4𝑟 + 1 =

𝑋 𝑘 + 𝑗𝑋 𝑘 +𝑁

4 − 𝑋 𝑘 +

𝑁

2 − 𝑗𝑋 𝑘 +

3𝑁

4 𝜔𝑁

−𝑘 𝑁/4 −1𝑘=0 𝜔𝑁/4

−𝑟𝑘 (3.40)

𝑥 4𝑟 + 2 =

𝑋 𝑘 − 𝑋 𝑘 +𝑁

4 + 𝑋 𝑘 +

𝑁

2 − 𝑋 𝑘 +

3𝑁

4 𝜔𝑁

−2𝑘 𝑁/4 −1𝑘=0 𝜔𝑁/4

−𝑟𝑘 (3.41)

𝑥 4𝑟 + 3 =

𝑋 𝑘 − 𝑗𝑋 𝑘 +𝑁

4 − 𝑋 𝑘 +

𝑁

2 + 𝑗𝑋 𝑘 +

3𝑁

4 𝜔𝑁

−3𝑘 𝑁/4 −1𝑘=0 𝜔𝑁/4

−𝑟𝑘 (3.42)

40

Each of the four equations listed above represent the Radix-4 IFFT and can be described

as four length 𝑁/4 IFFTs. As with any FFT, a butterfly chart can be used to visualize the

processing that occurs in order to compute an IFFT. As an example, consider a length

four Radix-4 IFFT as shown in Figure 3.5.

j

-j

-1

-1

-1

-j

-1

j

x(n)

x(n + N/4)

x(n + N/2)

x(n + 3N/4)

X(k)

X(k + N/4)

X(k + N/2)

X(k + 3N/4)

+

+

+

+

ω0Nω

nkN/4

ωnkN/4ωn

N

ωnkN/4ω2n

N

ωnkN/4ω3n

N

Fig. 3.5. Four Point Radix-4 IFFT Butterfly Diagram

Utilizing the concepts developed to simplify the FFT, the same approach can be

applied to the implementation of the Radix-4 IFFT algorithm. Reviewing each of the

four equations that comprise the Radix-4 IFFT, it can be observed that the only

multiplications are with respect to the twiddle factors associated with each 𝑁/4 IFFT.

Focusing specifically on Equation 3.39, the collective twiddle factor is represented as in

Equation 3.43.

𝑡𝑓1 = 𝜔𝑁−0𝜔𝑁/4

−𝑟𝑘 (3.43)

41

To start, 𝑡𝑓1 can be further reduced as shown in Equation 3.44.

𝑡𝑓1 = 𝜔𝑁/4−𝑟𝑘 (3.44)

Using the properties defined to create the rounded IDFT matrix, a quantized version of

𝑡𝑓1 can be defined as follows.

𝑡𝑓 1 = 𝑟𝑐𝑜𝑠𝑘 8𝜋𝑘𝑟

𝑁 + 𝑗 𝑟𝑠𝑖𝑛𝑘

8𝜋𝑘𝑟

𝑁 (3.45)

Twiddle factor 𝑡𝑓 1 can then be applied to Equation 3.39, resulting in a multiplier-less

IFFT as shown in Equation 3.46.

𝑥 4𝑟 = 𝑋 𝑘 + 𝑋 𝑘 +𝑁

4 + 𝑋 𝑘 +

𝑁

2 + 𝑋 𝑘 +

3𝑁

4

𝑁/4 −1𝑘=0 𝑡𝑓 1 (3.46)

The remaining three equations that comprise the Radix-4 are not as simply quantized.

Looking in detail with respect to Equation 3.40, the associated twiddle factor is defined

as in Equation 3.47.

𝑡𝑓2 = 𝜔𝑁−𝑘𝜔𝑁/4

−𝑟𝑘 (3.47)

Furthermore, twiddle factor 𝑡𝑓2 can be expanded as shown in Equation 3.48.

𝑡𝑓2 = 𝜔𝑁−𝑘𝜔𝑁/4

−𝑟𝑘 = 𝑒𝑗2𝑘𝜋

𝑁 𝑒𝑗8𝑟𝑘𝜋

𝑁 = 𝑒𝑗2𝑘𝜋

𝑁+

𝑗8𝑟𝑘𝜋

𝑁 = 𝑒 𝑗2𝑘𝜋

𝑁 1+4𝑟

(3.48)

Using the properties defined to create the rounded IDFT matrix, a quantized version of

𝑡𝑓2 can be defined as follows.

𝑡𝑓 2 = 𝑟𝑐𝑜𝑠𝑘 2𝑘𝜋

𝑁 1 + 4𝑟 + 𝑗 𝑟𝑠𝑖𝑛𝑘

2𝑘𝜋

𝑁 1 + 4𝑟 (3.49)

The definition of 𝑡𝑓 2 can be applied to Equation 3.40, resulting in the multiplier-less

sequence presented in Equation 3.50.

𝑥 4𝑟 + 1 = 𝑋 𝑘 + 𝑗𝑋 𝑘 +𝑁

4 − 𝑋 𝑘 +

𝑁

2 − 𝑗𝑋 𝑘 +

3𝑁

4

𝑁/4 −1𝑘=0 𝑡𝑓 2 (3.50)

42

Furthermore, twiddle factors 𝑡𝑓 3 and 𝑡𝑓 4 can be represented as shown in Equation 3.51

and Equation 3.52 and applied to define Equation 3.53 and Equation 3.54 in order to fully

represent the rounded IFFT.


𝑁 1 + 2𝑟 + 𝑗 𝑟𝑠𝑖𝑛𝑘

4𝑘𝜋

𝑁 1 + 2𝑟 (3.51)


𝑁 3 + 4𝑟 − 𝑗 𝑟𝑠𝑖𝑛𝑘

2𝑘𝜋

𝑁 3 + 4𝑟 (3.52)

𝑥 4𝑟 + 2 = 𝑋 𝑘 − 𝑋 𝑘 +𝑁

4 + 𝑋 𝑘 +

𝑁

2 − 𝑋 𝑘 +

3𝑁

4

𝑁/4 −1𝑘=0 𝑡𝑓 3 (3.53)

𝑥 4𝑟 + 3 = 𝑋 𝑘 − 𝑗𝑋 𝑘 +𝑁

4 − 𝑋 𝑘 +

𝑁

2 + 𝑗𝑋 𝑘 +

3𝑁

4 𝑡𝑓 4

𝑁/4 −1𝑘=0 (3.54)

To conclude the definition of the rounded IFFT, the butterfly diagram of Figure 3.5 has

been updated as shown in Figure 3.6, to clearly display the quantized twiddle factors.

j

-j

-1

-1

-1

-j

-1

j

x(n)

x(n + N/4)

x(n + N/2)

x(n + 3N/4)

X(k)

X(k + N/4)

X(k + N/2)

X(k + 3N/4)

+

+

+

+

Fig. 3.6. Four Point Rounded Radix-4 IFFT Butterfly Diagram

43

3.6 Simple SISO OFDM

Now that simplified versions of both the FFT and IFFT algorithms have been

defined, it is time to apply both to the SISO OFDM system. To start, consider the

simplified OFDM multi-carrier modulation system with single transmit and receive

antennas as illustrated in Figure 3.7.

Simp

IFFTP/S

X(0)

X(1)

X(N-1)

.

.

.

S/PSimp

FFT

Channel

Estimation

.

.

.

.

.

.

Y(0)

Y(1)

Y(N-1)

Add

CP

.

.

.

h +

w

Remove

CP

.

.

.

Simp

FFT

\

.

.

.

\

\

.

.

.

Fig. 3.7. Simplified SISO OFDM Transceiver Block Diagram

It is important to the note that the difference between Figure 2.2 and Figure 3.7 is specific

to the simplification of the FFT and IFFT blocks. The complex information symbols are

denoted in Figure 3.7 by 𝑋(𝑖), where 𝑖 = 0,1, … , 𝑁 − 1, where 𝑁 is the total number of

carriers. The values associated with the complex symbols are derived from bit-to-symbol

mapping techniques such as QPSK and QAM. The simplified IFFT block provides the

capability to transform complex information symbols, represented by 𝑿 into an

approximate time domain representation via the rounded IFFT algorithm. Depending on

the level of quantization, execution of the rounded IFFT algorithm provides near carrier

44

orthogonality, still allowing for successful OFDM communications. Error associated

with carrier orthogonality, is injected into the system due to the characteristics of the

simplified IFFT algorithm. As the level of quantization specified for the rounded IFFT

increases, errors due to approximate orthogonality reduce. In this particular description,

the rounded IFFT length is equal to the number of carriers associated with 𝑿, defined as

𝑁. In order to provide a mathematical representation of the rounded IFFT, the notation

𝑭 𝑵−1 is introduced in Equation 3.55 in order to represent the rounded IDFT matrix of size

𝑁𝑥𝑁.

𝒙 = 𝑭 𝑵−1𝑿 (3.55)

Vector 𝒙 is the result of evaluating Equation 3.55, which is the length 𝑁 approximate

time domain representation of 𝑿. Stepping through Figure 3.7 from left to right, a cyclic

prefix (CP) of length 𝐾 must be applied to vector 𝒙. In this description, the CP is

included in signal 𝒙 via the identical procedure defined for the conventional SISO OFDM

system, resulting in Equation 3.56.

𝒙 𝒄𝒑 =

𝑥 𝑁 − 𝐾 , 𝑥 𝑁 − 𝐾 − 2 , …𝑥 𝑁 − 1 , 𝑥 0 , 𝑥 1 , … , 𝑥 𝑁 − 1 (3.56)

As in the standard system, it is assumed that the channel can be characterized by slow

fading and thus the channel impulse response does not change within one OFDM symbol.

Once the CP is incorporated into 𝒙 , 𝒙 𝒄𝒑 is transmitted through the wireless channel. At

the receiver, signal 𝒚 is mathematically represented by the linear convolution between

transmitted signal 𝒙 𝒄𝒑 and the channel impulse response, plus channel noise 𝒘, as

specified in Equation 3.57 and Equation 3.58.

45

𝒚 𝒄𝒑 = 𝒙 𝒄𝒑 ∗ 𝑕𝑙 + 𝒘 (3.57)

𝑦 𝑐𝑝 𝑚 = 𝑕𝑙𝑥 𝑐𝑝 𝑚 − 𝑙 + 𝑤 𝑚 , 𝑚 = 0,1, …𝑁 + 𝐾 + 𝐿 − 2𝐿−1𝑙=0 (3.58)

Equation 3.57 generically describes the convolution, where as Equation 3.58 represents

the convolution by its mathematical definition. Channel noise 𝒘 is defined as Additive

White Gaussian Noise (AWGN) with zero mean, variance 𝜎2 =𝑁0

2 and 𝑁0 is the single-

sided power spectral density. In simple OFDM, the CP is removed from received signal

𝒚 𝒄𝒑 in the same manner as in the conventional system per Equation 3.59.

𝒚 = 𝒚 𝒄𝒑 𝐾 : (𝑁 + 𝐾 − 1) (3.59)

The removal of the CP converts the linear convolution between the transmitted symbols

and the channel impulse response into a cyclic convolution. The result of the cyclic

convolution is a wireless system defined in accordance with Equation 3.60.

𝒚 = 𝒉 𝒙 + 𝒘 (3.60)

Incorporating the relationship specified in Equation 3.55, signal 𝒚 can further be

expressed as defined in Equation 3.61.

𝒚 = 𝒉 𝑭 𝑵−𝟏𝑿 + 𝒘 (3.61)

The next step in the receive chain is to apply the rounded FFT to signal 𝒚 . Similar to the

system’s use of the rounded IFFT contained in the transmitter, the simple FFT length is

equal to 𝑁. In order to provide a mathematical representation of the rounded FFT, the

notation 𝑭 𝑵 is introduced in Equation 3.62 to represent the rounded Discrete Fourier

Transform (DFT) matrix of size 𝑁𝑥𝑁.

𝒀 = 𝑭 𝑵𝒚 (3.62)

46

The relationship defined in Equation 3.62 can be substituted into Equation 3.61 to define

Equation 3.63.

𝒀 = 𝑭 𝑵𝒉 𝑭 𝑵−1𝑿 + 𝑭 𝑵𝒘 (3.63)

In order for the relationships defined in this section to successfully represent a

communication system, the information symbols that originated as 𝑿 must be recovered

from 𝒀 through equalization. Equalization can be accomplished by multiplying the

inverse of the relationship 𝑭 𝑵𝒉 𝑭 𝑵−1 with 𝒀 . Due to the quantization utilized in the

development of matrix 𝑭 𝑵 and 𝑭 𝑵−1, the resultant matrix formed by the relationship

𝑭 𝑵𝒉 𝑭 𝑵−1 will not result in a completely diagonal matrix as in the conventional system.

As such, the processing necessary to compute 𝑭 𝑵𝒉 𝑭 𝑵−1 is more complicated than in

standard OFDM. As a result, an estimate of the transmitted information 𝑿 can still be

obtained via multiplication of the inverse matrix of 𝑭 𝑵𝒉 𝑭 𝑵−1 with 𝒀, however a simple

zero-forcing equalizer is preferred as it provides identical complexity with respect to the

standard system.

𝑿 = 𝑭 𝑵𝒉 𝑭 𝑵−1

−1𝒀 (3.64)

A simple zero-forcing (ZF) detector, that requires one division per carrier as defined in

Equation 3.65 can be implemented to determine an estimate of 𝑿.

𝑋 𝑀 = 𝑌 𝑀

𝐻 𝑀 𝑤𝑕𝑒𝑟𝑒 𝑀 = 0,1, … , 𝑁 − 1 (3.65)

Figures 4.1 through 4.27, contained in Chapter 4, provide the BER curves

necessary to evaluate the performance of the conventional SISO OFDM system described

in Chapter 2.2 with respect to the simplified SISO OFDM system.

47

3.7 Simple MIMO OFDM

In order to study the application of the rounded FFT and IFFT algorithms to

MIMO OFDM multi-carrier modulation, consider a system with 𝑚 transmit and 𝑛 receive

antennas as illustrated in Figure 3.8.

Simp

IFFTP/S

X1(0)

X1(1)

X1(N-1)

.

.

.

.

.

.

S/P Simp FFT.

.

.

VBLAST

Symbol

Detection

.

.

.

.

.

.

Y1(0)

Y1(1)

Y1(N-1)

Simp

IFFT

Xm(0)

Xm(1)

Xm(N-1)

.

.

.

.

.

.

Yn(0)

Yn(1)

Yn(N-1)

Add

CP

h11 +

Remove

CP

P/S.

.

.

S/P Simp FFT.

.

.

.

.

.

Add

CP

hnm +

Remove

CP

hn1 +

h1m +

w

w

Fig. 3.8. Simplified V-Blast MIMO OFDM Receiver/Transmitter Block Diagram

In this system, the complex information symbols associated with the 𝑖𝑡𝑕 transmitter are

identified as 𝑿𝒊. Each set of complex symbols 𝑿𝒊 is derived from a data set defined as 𝑿.

For example, if the number of transmitters is equal to three, then 𝑚 equals three and 𝑿

would have a vector length 3𝑁 where 𝑁 is the total number of carriers associated with a

single transmitter. In order to define each 𝑿𝒊, 𝑿 is parsed into 𝑚 data vectors of equal

length such that different sets of complex symbols can be transmitted in parallel. The

values associated with the complex symbols of 𝑿 are derived from bit-to-symbol

mapping techniques such as QPSK and QAM. Similar to the SISO case, the rounded

48

IFFT blocks represented in Figure 3.8 provide the capability to transform the complex

information symbols associated with a specific transmitter, into a near time domain

representation. The evaluation of the rounded IFFT algorithm provides near

orthogonality between carriers specific to a transmitter such that successful OFDM

communications can still be obtained. It is important to note that as the quantization level

associated with the rounded IFFT increases, errors due to approximate orthogonality are

reduced. Each rounded IFFT is of length 𝑁, which is equal to the number of carriers

associated with each 𝑿𝒊. In order to provide a mathematical representation of the

rounded IFFT with respect to 𝑚 number of transmitters, the notation 𝑭 𝑵−1 is introduced in

Equation 3.66 to represent the rounded Inverse Discrete Fourier Transform (IDFT) matrix

of size 𝑁𝑥𝑁.

𝒙 = (𝑭 𝑵−1⨂𝑰𝒎)𝑿 (3.66)

Progressing through Figure 3.8 from left to right, a cyclic prefix (CP) of length 𝐾 must be

applied to each vector 𝒙 𝒊. In this description of the simplified MIMO OFDM system, the

CP is applied to each 𝒙 𝒊 in the same manner as in the conventional system with the result

indicated in Equation 3.67.

𝒙 𝒄𝒑𝒊𝑇 =

𝑥 𝑖 𝑁 − 𝐾 , 𝑥 𝑖 𝑁 − 𝐾 + 1 , …𝑥 𝑖 𝑁 − 1 , 𝑥 𝑖 0 , 𝑥 𝑖 1 , … , 𝑥 𝑖 𝑁 − 1 (3.67)

The MIMO wireless channel can be modeled as a matrix of channel coefficients in

accordance with every possible combination of transmit and receive antennas. Equation

3.68 provides a generic representation of the MIMO channel for 𝑚 transmitters and 𝑛

receivers. Each specific channel coefficient 𝑕𝑗𝑖 , where 𝑗 identifies the receiver and 𝑖

49

identifies the transmitter, is a complex Gaussian random variable that provides the fading

gain between each variation of transmitter and receiver data path.

𝒉 =

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

(3.68)

As in previous descriptions, it is assumed that the MIMO channel can be characterized by

slow fading and thus the channel impulse response does not change within one OFDM

symbol. This analysis also does not account for the multi-path associated with each

specific combination of transmit and receive antenna and thus no CP is actually required.

In general though, the same principles used to define the length of the CP for the

previously discussed systems also apply to simple MIMO. Once 𝒙 𝒄𝒑 is defined as

indicated in Equation 3.67, the corresponding data is transmitted through the MIMO

wireless channel. At each receiver in the simple MIMO system, the CP associated with

received signal 𝒚 𝒄𝒑 is discarded as indicated by Equation 3.69.

𝒚 =

𝒚 𝒄𝒑𝟏 𝐾 − 1 : (𝑁 + 𝐾 − 1)

⋮𝒚 𝒄𝒑𝒎 𝐾 − 1 : (𝑁 + 𝐾 − 1)

(3.69)

Once the CP is removed, signal 𝒚 can be mathematically represented as the linear

convolution between the transmitted signal 𝒙 and associated MIMO channel coefficient,

plus channel noise 𝒘, as specified in Equation 3.70.

𝒚 𝟏

𝑇

𝒚 𝟐𝑇

⋮𝒚 𝒏

𝑇

=

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

∗

𝒙 𝟏

𝑇

𝒙 𝟐𝑇

⋮𝒙 𝒎

𝑇

+

𝒘𝟏

𝑇

𝒘𝟐𝑇

⋮𝒘𝒏

𝑇

(3.70)

50

Channel noise 𝑤 included in Equation 3.70, exists for each spatial path and is Additive

White Gaussian Noise (AWGN) with zero mean and variance 𝜎2 =𝑁0

2, where 𝑁0 is the

single-sided power spectral density. The channel noise associated with the 𝑗𝑡𝑕 receive

antenna can be represented as follows.

𝒘𝒋 =

𝑤𝑗 0

𝑤𝑗 1

⋮𝑤𝑗 (𝑁 − 1)

(3.71)

At this point, the rounded DFT of length 𝑁 is applied to received signal 𝒚 in order to

reverse the impact of the rounded IDFT modulation in the transmitter. In order to

provide a mathematical representation of the rounded DFT, the notation 𝑭 𝑵 is introduced

in Equation 3.72.

𝒀 = (𝑭 𝑵⨂𝑰𝒏)𝒚 (3.72)

After computing 𝒀 , the next step in the receiver is to determine an approach for

recovering an estimate of 𝑿 from 𝒀 . Analyzing the assumptions made with respect to the

simplified MIMO OFDM system, it can be concluded that the linear convolution between

the MIMO channel matrix 𝒉 and transmitted signal 𝒙, as indicated in Equation 3.70, can

be rewritten as follows with multiplication due to the singular duration of channel

impulse response.

𝒚 𝟏

𝒚 𝟐

⋮𝒚 𝒏

=

𝑕11𝒙 𝟏 + 𝑕12𝒙 𝟐 + ⋯ + 𝑕1𝑚𝒙 𝒎

𝑕21𝒙 𝟏 + 𝑕22𝒙 𝟐 + ⋯ + 𝑕2𝑚𝒙 𝒎

⋮𝑕𝑛1𝒙 𝟏 + 𝑕𝑛2𝒙 𝟐 + ⋯ + 𝑕𝑛𝑚 𝒙 𝒎

+

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(3.73)

With this arrangement, the relationship expressed in Equation 3.72 can be substituted into

Equation 3.73 as follows.

51

𝑌 1𝑌 2

⋮𝑌 𝑛

= (𝑭 𝑵⨂𝑰𝒏)

𝑕11𝒙 𝟏 + 𝑕12𝒙 𝟐 + ⋯ + 𝑕1𝑚𝒙 𝒎 + 𝒘𝟏

𝑕21𝒙 𝟏 + 𝑕22𝒙 𝟐 + ⋯ + 𝑕2𝑚𝒙 𝒎 + 𝒘𝟐

⋮𝑕𝑛1𝒙 𝟏 + 𝑕𝑛2𝒙 𝟐 + ⋯ + 𝑕𝑛𝑚 𝒙 𝒎 + 𝒘𝒏

(3.74)

Furthermore, each signal 𝒙 𝒊 is approximately equal to the rounded IDFT of

corresponding vector 𝑿𝒊 and as such, can be utilized as specified in Equation 3.75.

𝒀 𝟏

𝒀 𝟐

⋮𝒀 𝒏

=

(𝑭 𝑵⨂𝑰𝒏)

𝑕11(𝑭 𝑵

−𝟏𝑿𝟏) + 𝑕12(𝑭 𝑵−𝟏𝑿𝟐) + ⋯ + 𝑕1𝑚 (𝑭 𝑵

−𝟏𝑿𝒎) + 𝒘𝟏

𝑕21(𝑭 𝑵−𝟏𝑿𝟏) + 𝑕22(𝑭 𝑵

−𝟏𝑿𝟐) + ⋯ + 𝑕2𝑚 (𝑭 𝑵−𝟏𝑿𝒎) + 𝒘𝟐

⋮𝑕𝑛1(𝑭 𝑵

−𝟏𝑿𝟏) + 𝑕𝑛2(𝑭 𝑵−𝟏𝑿𝟐) + ⋯ + 𝑕𝑛𝑚 (𝑭 𝑵

−𝟏𝑿𝒎) + 𝒘𝒏

(3.75)

Using the properties of the matrix formed by the Kronecker Product, the rounded DFT

matrix 𝐹 𝑁 can be transitioned into Equation 3.75 as follows.

𝒀 𝟏

𝒀 𝟐

⋮𝒀 𝒏

=

𝑕11(𝑭 𝑵𝑭 𝑵

−𝟏𝑿𝟏) + 𝑕12(𝑭 𝑵𝑭 𝑵−𝟏𝑿𝟐) + ⋯ + 𝑕1𝑚 (𝑭 𝑵𝑭 𝑵

−𝟏𝑿𝒎) + (𝑭 𝑵𝒘𝟏)

𝑕21(𝑭 𝑵𝑭 𝑵−𝟏𝑿𝟏) + 𝑕22(𝑭 𝑵𝑭 𝑵

−𝟏𝑿𝟐) + ⋯ + 𝑕2𝑚 (𝑭 𝑵𝑭 𝑵−𝟏𝑿𝒎) + (𝑭 𝑵𝒘𝟐)

⋮𝑕𝑛1(𝑭 𝑵𝑭 𝑵

−𝟏𝑿𝟏) + 𝑕𝑛2(𝑭 𝑵𝑭 𝑵−𝟏𝑿𝟐) + ⋯ + 𝑕𝑛𝑚 (𝑭 𝑵𝑭 𝑵

−𝟏𝑿𝒎) + (𝑭 𝑵𝒘𝒏)

(3.76)

Additional reduction can be performed given the fact that a matrix multiplied by its

inverse results in an Identity matrix. Utilizing said property, Equation 3.76 can be

simplified to the following.

𝒀 𝟏

𝒀 𝟐

⋮𝒀 𝒏

=

𝑕11𝑿𝟏 + 𝑕12𝑿𝟐 + ⋯ + 𝑕1𝑚𝑿𝒎

𝑕21𝑿𝟏 + 𝑕22𝑿𝟐 + ⋯ + 𝑕2𝑚𝑿𝒎

⋮𝑕𝑛1𝑿𝟏 + 𝑕𝑛2𝑿𝟐 + ⋯ + 𝑕𝑛𝑚𝑿𝒎

+ (𝑭 𝑵⨂𝑰𝒏)

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(3.77)

52

At this point, symbol 𝑾 is defined as follows to represent the frequency representation of

AWGN.

𝑾 =

𝑾 𝟏

𝑾 𝟐

⋮𝑾 𝒏

= (𝑭 𝑵⨂𝑰𝒏)

𝒘𝟏

𝒘𝟐

⋮𝒘𝒏

(3.78)

For the purposes of simplicity going forward with the explanation, vectors 𝒀 , 𝑿 and 𝑾

are redefined as follows.

𝒀 =

𝒀 𝟏

𝑇

𝒀 𝟐𝑇

⋮𝒀 𝒏

𝑇

, 𝑿 =

𝑿𝟏

𝑇

𝑿𝟐𝑇

⋮𝑿𝒎

𝑇

, 𝑾 =

𝑾 𝟏

𝑇

𝑾 𝟐𝑇

⋮𝑾 𝒏

𝑇

(3.79)

With the definition of Equation 3.79, the simple MIMO OFDM communication system

can be represented in a format easily extended for additional processing as shown in

Equation 3.80.

𝒀 =

𝒀 𝟏

𝑇

𝒀 𝟐𝑇

⋮𝒀 𝒏

𝑇

=

𝑕11 𝑕12

𝑕21 𝑕22

… 𝑕1𝑚

… 𝑕2𝑚

⋮ ⋮𝑕𝑛1 𝑕𝑛2

… ⋮… 𝑕𝑛𝑚

𝑿𝟏

𝑇

𝑿𝟐𝑇

⋮𝑿𝒎

𝑇

+

𝑾 𝟏

𝑇

𝑾 𝟐𝑇

⋮𝑾 𝒏

𝑇

= 𝒉𝑿 + 𝑾 (3.80)

With the specification of Equation 3.80, the communications model has been defined to

the point where an estimate of 𝑿 can be determined using a standard V-Blast approach.

All V-Blast approaches introduced for the conventional MIMO OFDM system also apply

to the simplified system. Once an estimate of 𝑿 is determined using a standard V-Blast

technique, the BER is computed in order to evaluate the communication system

performance.

53

Figure 4.28 through Figure 4.43, contained in Chapter 4, provide the BER curves

necessary to compare the performance of the conventional MIMO OFDM system

described in Chapter 2.4 with the simplified MIMO OFDM system.

54

4. SIMULATION RESULTS

4.1 SISO OFDM Architecture

Using the relationships defined in Chapters 2 and 3, a MATLAB model has been

developed in order to represent the performance of the conventional and simplified SISO

OFDM modulated systems [1]. Simulations of the model have been executed to generate

the results presented in this section. The key parameter used to analyze the performance

of the system is bit error rate (BER). As such, BER curves with respect to the ratio of bit

energy to single-sided noise power spectral density are computed in order to evaluate the

performance of the conventional system with respect to the simplified system. With

regards to the model developed to represent the simplified system, the levels of

quantization used to implement the rounded FFT and rounded IFFT are as follows.

Table 4.1

Rounded FFT/IFFT Twiddle Factor Quantization

𝑘 Quantization

Steps

2 5

4 9

8 17

16 33

55

Other degrees of freedom considered in the simulations are bit-to-symbol mappings of

QPSK and 16QAM as well as three different wireless channel models. The first channel

model represents a flat fading channel with the following frequency response.

Fig. 4.1. Flat Fading Channel Frequency Response (Channel 1)

The flat fading channel will attenuate the magnitude of the transmission by slightly less

than -3 dB; however, there is no effect on the transmission phase. The second channel

model included in this analysis represents a typical office environment with 50 nano

second root mean square (RMS) delay spread and Rayleigh fading. The frequency

response of the second channel is as follows.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Normalized Frequency ( rad/sample)

Phase (

degre

es)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2


Magnitude (

dB

)

56

Fig. 4.2. Typical Office Channel Frequency Response (Channel 2)

The last channel included in this research characterizes a large open area with 100 nano

second RMS delay spread and Rayleigh fading. The frequency response of channel

number three is as follows.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200


Phase (

degre

es)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-15

-10

-5

0


Magnitude (

dB

)

57

Fig. 4.3. Large Open Area Channel Frequency Response (Channel 3)

In conjunction with the degrees of freedom included in the SISO OFDM model, static

parameters such as a symbol rate equal to 250 KHz, presence of AWGN and FFT/IFFT

and rounded FFT/IFFT Length of 64 are employed. The following figures are plots

generated to describe the performance of the conventional SISO OFDM system versus

the rounded SISO OFDM system.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-100

0

100

200


Phase (

degre

es)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-20

-10

0

10


Magnitude (

dB

)

58

Fig. 4.4. SISO OFDM with QPSK BER, k=2, Channel 1


5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

59



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

60

Fig. 4.8. SISO OFDM with 16QAM BER, k=2, Channel 1


5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

61



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

62



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

63



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

64



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

65



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

66



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

67



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM QPSK

Rounded OFDM QPSK

68



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

69



5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

OFDM 16QAM

Rounded OFDM 16QAM

70

4.2 MIMO OFDM Architecture

Similar to the SISO system, a computer simulation has been developed in

MATLAB to provide the performance of both the conventional and simplified MIMO

OFDM modulated systems [1]. Simulations of the MIMO model have been performed in

order to generate the results presented in this section. As in the SISO system, the key

parameter used to analyze the performance of the MIMO architecture is BER. As such,

several BER curves are computed in order to evaluate the performance of the

conventional system with respect to the simplified system. The levels of quantization

used to implement the rounded FFT and rounded IFFT for the simplified MIMO

architecture are the same as in the simulations for the simplified SISO model. The

quantization levels used to execute the rounded FFT and rounded IFFT are indicated in

Table 4.1. Other parameters included in this analysis are bit-to-symbol mappings of

QPSK and 16QAM as well as the symbol detection technique of Optimal Ordered SIC

coupled with ZF and MMSE equalization. Additionally, results are generated for

randomly generated flat fading complex channel. Static parameters used in the

simulation are a symbol rate equal to 250 KHz, presence of AWGN, FFT/IFFT and

rounded FFT/IFFT Length of 64, two transmit antennas and two receive antennas. The

following is a series of plots to describe the performance of the conventional SISO

OFDM system versus the rounded SISO OFDM system.

71

Fig. 4.28. MIMO OFDM with Optimal Ordered ZF-SIC and QPSK BER, k=2


0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

72



0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

73

Fig. 4.32. MIMO OFDM with Optimal Ordered MMSE-SIC and QPSK BER, k=2


0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

74



0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

75

Fig. 4.36. MIMO OFDM with Optimal Ordered ZF-SIC and 16QAM BER, k=2


0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

76



0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

77

Fig. 4.40. MIMO OFDM with Optimal Ordered MMSE-SIC and 16QAM BER, k=2


0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

78



0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

Bit E

rror

Rate

Eb/N0

Rounded MIMO-OFDM

MIMO-OFDM

79

5. CONCLUSIONS

A new concept has been developed that can be utilized in conjunction with

existing systems such that they can be considered cutting-edge and innovative. The new

concept introduced in this research is the application of the rounded Fast Fourier

Transform (FFT) and rounded Inverse Fast Fourier Transform (IFFT) to both SISO and

MIMO OFDM modulated systems. As the descriptions and associated results presented

in this research confirm, inclusion of the rounded FFT and rounded IFFT into both SISO

and MIMO OFDM systems provide performance that approaches the conventional

system, while eliminating all non-trivial multiplications. Furthermore, the results prove

that the approach introduced for simple OFDM leads to viable communication systems.

Considering the results associated with the simple SISO architecture presented in

Chapter 4.1, it is clear that the performance of systems that include QPSK bit-to-symbol

mappings as opposed to QAM, more closely resemble the performance of the

conventional. It is also evident that the low end of “twiddle factor” quantization (i.e. k =

2, k = 4) performs poorly in the SISO system as error is introduced both in the multi-

carrier modulation process as well as in the equalizer due to the use of the rounded FFT

to generate a frequency domain representation of the channel impulse response. With

regards to the MIMO system, the simulation results presented in Chapter 4.2 clearly

indicate that the performance degradation is smaller than what has been concluded for the

80

SISO system. This observation is significant for systems that implement MIMO as the

IFFT and FFT algorithms are required for every spatial stream (i.e. antenna) and thus the

system-wide computational complexity is further reduced. Therefore, the proposed

approach is particularly suitable for modern high-data rate MIMO systems.

The elimination of non-trivial multiplications provided by the rounded FFT and

IFFT will allow for simpler hardware implementation due to the reduction in

computational complexity. The reduction in computational complexity is quantified by

comparing the total number of actual multiplications and additions necessary to

implement the conventional Radix-2 and Radix-4 FFT versus the rounded algorithm as

described by Table 5.1.

Table 5.1

FFT and Rounded FFT Complexity

Transform

Size

Multiplications Additions

Radix-2

FFT

Radix-4

FFT

Rounded

FFT

Radix-2

FFT

Radix-4

FFT

Rounded

FFT

64 264 208 k = 2: 0 1032 976 k = 2: 976

k = 4: 0 k = 4: 1008

k = 8: 0 k = 8: 1032

k = 16: 0 k = 16: 1072

256 1800 1392 k = 2: 0 5896 5488 k = 2: 5488

k = 4: 0 k = 4: 5616

k = 8: 0 k = 8: 5736

k = 16: 0 k = 16: 5844

The values provided for additions and multiplications as represented in Table 5.1 are the

actual number of non-trivial real multiplications and real additions. As previously

described, different levels of quantization can be utilized to develop the rounded FFT and

IFFT. As the level of quantization increases, the overall system performance approaches

81

that of the conventional system. Depending on the performance required by a specific

application, different variations of the rounded FFT and IFFT can be utilized in order to

obtain minimum complexity.

Table 5.1 clearly describes that the implementation rounded FFT requires a slight

increase of additions along with zero non-trivial multiplications when compared to the

Radix-2 and Radix-4 FFT; however, this result must be further quantified. Using the

comparison of additive complexity versus multiplicative complexity provided in

Equation 3.13, a ratio can be developed in order to scale the complexity associated with

the implementation of a multiplication to be consistent with an addition [11].

Furthermore, an estimate of overall complexity can be computed and compared for the

Radix-2 FFT, Radix-4 FFT and rounded FFT with the results contained in Table 5.2.

Table 5.2

Complexity Reduction Provided by Rounded FFT

Number

of Bits

Order of

Multiplicative

Complexity

vs. Additive

Complexity

Estimated Complexity Reduction in

Complexity

256

Length

Radix-2

FFT

256

Length

Radix-4

FFT

256 Length

Rounded

FFT (k=16)

Rounded

FFT vs.

Radix-2

Rounded

FFT vs.

Radix-4

7 2.65 10666 9177 5844 45% 36%

8 2.83 10990 9427 5844 47% 38%

9 3.00 11296 9664 5844 48% 40%

10 3.16 11584 9887 5844 50% 41%

11 3.32 11872 10109 5844 51% 42%

12 3.46 12124 10304 5844 52% 43%

13 3.61 12394 10513 5844 53% 44%

14 3.74 12628 10694 5844 54% 45%

15 3.87 12862 10875 5844 55% 46%

16 4.00 13096 11056 5844 55% 47%

82

To summarize the results included in Table 5.2, the rounded FFT provides an estimated

minimum reduction in complexity of 45% with respect to the Radix-2 FFT and a

minimum reduction in complexity of 36% with respect to the Radix-4 FFT for systems

that represent numerical values with a number of bits greater than six. Table 5.2 provides

the critical result necessary to quantify the reduction in complexity for the new rounded

system.

BIBLIOGRAPHY

83

BIBLIOGRAPHY

[1] H. Harada and R. Prasad, Simulation and Software Radio for Mobile

Communications. Boston: Artech House, 2002.

[2] J. Bingham, “Multicarrier modulation for data transmission: An idea whose time has

come,” IEEE Commun. Mag., pp. 5-14, May 1990.

[3] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-division

multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Tech., vol.

COM-19, no. 10 pp. 628-634.

[4] T. Cooklev and P. Siohan, “Vector transform-based OFDM,” Proc. Asilomar Conf.

Communications and Signal Processing, Pacific Grove, CA, Nov. 2006.

[5] P.W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST:

An architecture for realizing very high data rates over the rich-scattering wireless

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A Simplified Approach to Multi-carrier Modulation

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