Multichannel and Multicarrier Systems · • This chapter covers multichannel and multicarrier...

Multichannel and Multicarrier Systems PD0192F Digital Communications

JOÃO VIEIRA

Outline

• Multichannel digital Communications in AWGN channels;

• Multicarrier communications:

– SC vs MC;

– Capacity of Nonideal linear channels;

– OFDM – modulation and demodulation;

– FFT implementation;

– Spectral characteristics;

– Bit and power allocation;

– Peak-to-Average Ratio

Quick Intro

• In some applications, its desirable to transmit the same signal in

different channels is prefered when using unreliable channels w/

high probability;

• Same information in multiple channels diversity;

• Another form of multichannel comm. is multicarrier transmission,

where a freq. band is subdivided into a number of subchannels (and

information is transmitted in each of the subchannels);

• This chapter covers multichannel and multicarrier signal

transmission;

• Here we focus on AWGN channels (multi-carrier/channel in fading

channels is addressed in Chapter 13).

11.1- Multichannel Digital

communications in AWGN channels

Channels are fixed and only differ in magnitude and phase;

Waveforms have equal energy and are equally likely;

N. channels N. waveforms



Decision

variables

For coherent detection:



For non-coherent detection:

For binary signaling and assuming that

are the transmitted waveforms. An error is commited if:

or

Square law

Detection

(easier to analyse)

No attempt is made to estimate the channel parameters;

Detector decision is based on the sum of envelopes or sum of

squared envelopes (square law detection) of the MF outputs;






or


11.1-1 Binary Signals

The error probability for binary multichannel signalling in

AWGN is derived in Appendix B.

– Antipodal signals + perfect estimation of

– Rx combines energy of L channels such that no loss/gain in performance

when dividing Tx signal energy in channels;

• This holds if . If the estimates are not perfect, a loss in performance

occurs (Appendix C).

SNR per bit

If , ∀ n

Total Tx signal energy for the L signals.


Perfect CSI is an extreme case. On the other extreme DPSK;

“In differentially encoded PSK, the information sequence determines the relative phase, or

phase transition, between adjacent symbol intervals”

“Key assumption … phase offset φ remains the same over adjacent signaling periods”


Perfect CSI is an extreme case. On the other extreme DPSK;

If , for the same SNR/bit the multichannel performance

poorer than single-channel system splitting energy performance loss;

, ∀ n

“In differentially encoded PSK, the information sequence determines the relative phase, or

phase transition, between adjacent symbol intervals”


• A loss in performance also occurs in square law detection of orthogonal signals

transmitted in L channels; For binary orthogonal signalling Pe is identical to DPSK

but 3dB poorer ,i.e., 𝛾𝑏 is replaced by 𝛾𝑏

2 and the peformance loss due to

non-coherent combination of the L received signals is identical.

11.2 Multicarrier Communications

• (One) Motivation: reduce ISI and performance degradation

compared to SC systems.

11.2 Single-Carrier Vs Multicarrier

modulation

• If and are constant, the channel is ideal or

nondistorting;

• If we use single carrier on nonideal linear channels ISI since

the time dispersion >> symbol rate equalizer is necessary;

• Alternatively, we can divide the BW into almost ideal channels:

Let be the frequency response of a nonideal, band-limited channel

~ constant within

each sub-band

Constrain

11.2-2 – Capacity of a Nonideal Linear

Filter Channel

• Recall : (for ideal, band-limited AWGN)

• In multicarrier system

• When

• Under the previous constaint, the choice of P(f) that maximizes C

is choosen by maximizing:

11.2-2 – Capacity of a Nonideal Linear

Filter Channel

• The optimum distribution of Tx power:

• Basic interpretation: High power @ High SNR

11.2-3 – Orthogonal Frequency

Division Multiplexing (OFDM)

• Dividing channel into small sub-bands

can yield TX rates close to channel capacity

• Independently coded and modulated;

• Flat for each sub band -> no ISI.

• For each subchannel:

• If Δ𝑓 =1

𝑇

• OFDM is a special case of multichnnel mod. where subcarrier

acting in subchannel are mutually orthogonal.

11.2-4 – Modulation and demodulation

in an OFDM system

• With N large 𝑇 ≫ 𝜏𝑚𝑎𝑥 hence, fixed

• If QAM in each subcarrier, the signal @ k^th subcarrier can

be expressed as:

QAM signal point

In k^th subcarrier

symbol time in a SC system


in an OFDM system

• The received signal in the k^th subchannel is

• We assume 𝐶𝑘 and ϕ𝑘 known @ Rx;


in an OFDM system

• Demodulation on the kth subcarrier cross correlating 𝑟𝑘(𝑡) with the 2

basis, given that we know {ϕ𝑘}:

• and sampling the output of the cross-correlators at t = T, we obtain:

• Expressed as complex number:

• Scaling the received sybol by channel gain

Passed to detector


in an OFDM system

• Demodulation on the kth subcarrier cross correlating 𝑟𝑘(𝑡) with the 2

basis, given that we know {ϕ𝑘}:

• and sampling the output of the cross-correlators at t = T, we obtain:

From this description, 2 correlators (or 2 matched filters) per subcarrier to

demodulate the Rx signal. In an OFDM system 2N correlators;

This bank of 2N filters is equiv. to computation of DFT and its

inverse. Hence, FFT is an efficient tool to implement this system.

11.2-5 – An FFT Algorithm

Implementation of an OFDM system

Computation of IDFT of the data {X_k}, can be viewed as multiplication

of each data point X_k by a corresponding vector:



• Avoid ISI append cyclic prefix. (Re-indexing from –v to N-1);

• Conv. of with produces

• We are interested in thus the first v

samples can be discarded;

• Note: OFDM is not optimal;



• Due to cyclic prefix successive block frames do not interfere demodulated

sequence may be expressed as:

• When 𝑁 >> 𝑣, the rate loss due to cyclic prefix can be negligible.

• The transmission rate in each subchannel can be optimized by allocating the

average Tx power and number of bits. The SNR in the k^th subchannel is

defined as:

• Based on the SNR measurements, capacity may be determined as in

Section 11.2-2.

From a frequency-domain viewpoint, when the CIR is {𝑐𝑛, 0 ≤ n ≤ ν}, its F.R. at the

subcarrier frequencies 𝑓𝑘 = k/N is

11.2-6 – Spectral Characteristics of

Multicarrier systems

• OFDM subcarrier signals in are mutually orthogonal in time, they overlap in

frequency. This can be seen by computing the F.T. of the signal

• Spectral overlap of the main lobes but

signals are still orthogonal when transmitted

synchronously;

Sidenote: Doopler destroys orthogonality

among subcarriers;

11.2-7 – Bit and Power allocation in

Multicarrier Modulation

• … to optimize performance of MC system operatnig in linear-time AWGN channels;

• We assume Ñ subcarriers using QAM with 𝑀𝑖 = 2𝑏𝑖, and 𝑏𝑖 bits are transmitted in

the i^th subcarrier during T seconds.

• We assume complex constant 𝐶(𝑓𝑖) and flat noise PSD (across subchannels);

• Objective maximize Rb for a given Pe with is the same across the Ñ

subchannels (subject to a power constaint);

Error prob. For QAM

@ low error rates

total

bitrate

11.2-8 – Peak-to-Average Ratio in


• A major problem in MC modulation is the peak-to-average-power-ratio (PAPR);

occurs where many subchannels add constructively in phase;

Diff notation

Problem: the whole transmitter chain

(from baseband to antenna) needs to

have a linear/ high dynamic range.



• Clipping of the signal voltage in a D/A can happen, or it can saturate the power

amplifier in this, cause intermodulation distortion;

PAPR reducing methods:

• Tone-reservation: add a peak-annihilating signal c(n) to the transmit s(n). c(n)

is chosen such that it has counter-peaks peaks of s(n)+c(n) are lower than the

ones of s(n) .

Subcarriers need to be sacrificed;

• The desired “good” peak-annihilating property can be formulated as minimisation

problem:

average power per

signal sample



PAPR reducing methods:

• Tone-reservation;

• Tone-injection: same idea as tone-reservation (adding a peak-annihilating

signal); However, no tones are reserved: peak-annihilating signals are added on data-

carrying tones.

• Based on module principle;

• Rx has to know that the Tx may chose

to send points from the extended constellation;

• Selecting symbols from the extended

constellation increases the average Tx power

• (Connected with Tomlinson-Harashima

precoding in Section 9.5-4 ?;)

Multichannel and Multicarrier Systems · • This chapter covers multichannel and multicarrier...

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