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Chapter 1Complex Baseband Representation of Bandpass Signals

1.1 IntroductionAlmost every communication system operates by modulating an information bearing waveformonto a sinusoidal carrier. As examples, Table 1.1 lists the carrier frequencies of various methodsof electronic communication.

Type of Transmission Center Frequency of TransmissionTelephone Modems 1600-1800 Hz

AM radio 530-1600 KHzCB radio 27 MHzFM radio 88-108 MHz

VHF TV 178-216 MHzCellular radio 850 MHzIndoor Wireless Networks 1.8GHz

Commercial Satellite Downlink 3.7-4.2 GHzCommercial Satellite Uplink 5.9-6.4 GHz

Fiber Optics 2 1014 Hz

Table 1.1: Carrier frequency assignments for different methods of information transmission.

One can see by examining Table 1.1 that the carrier frequency of the transmitted signal

is not the component which contains the information. Instead it is the signal modulated onthe carrier which contains the information. Hence a method of characterizing a communicationsignal which is independent of the carrier frequency is desired. This has led communicationsystem engineers to use a complex baseband representation of communication signals tosimplify their job. All of the communication systems mentioned in Table 1.1 can be and typicallyare analyzed with this complex baseband representation. This handout develops the complexbaseband representation for deterministic signals. Other references which do a good job of developing these topics are [Pro89, PS94, Hay83, BB99]. One advantage of the complex baseband

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f C f C

W

G f qc ( )

4 Complex Baseband Representation of Bandpass Signals

representation is simplicity. All signals are lowpass signals and the fundamental ideas behindmodulation and communication signal processing are easily developed. Also any receiver thatprocesses the received waveform digitally uses the complex baseband representation to developthe baseband processing algorithms.

1.2 Baseband Representation of Bandpass SignalsThe rst step in the development of a complex baseband representation is to dene a bandpasssignal.

Denition 1.1 A bandpass signal, xc(t), is a signal whose one-sided energy spectrum is both:1) centered at a non-zero frequency, f C , and 2) does not extend in frequency to zero (DC).

The two sided transmission bandwidth of a signal is typically denoted by BT Hertz so thatthe one-sided spectrum of the bandpass signal is zero except in [ f C BT / 2, f C + BT / 2]. Thisimplies that a bandpass signal satises the following constraint: BT / 2 < f C . Fig. 1.1 shows atypical bandpass spectrum. Since a bandpass signal, xc(t), is a physically realizable signal it isreal valued and consequently the energy spectrum will always be symmetric around f = 0. Therelative sizes of BT and f C are not important, only that the spectrum takes negligible valuesaround DC. In telephone modem communications this region of negligible spectral values is onlyabout 300Hz while in satellite communications it can be many Gigahertz.

Figure 1.1: Energy spectrum of a bandpass signal.

A bandpass signal has a representation of

xc(t) = xI (t) 2 cos(2f ct) xQ (t) 2 sin(2f ct) (1.1)= xA (t) 2cos(2f ct + xP (t)) (1.2)

where f c is denoted the carrier frequency with f C BT / 2 f c f C + BT / 2. The signal xI (t) in(1.1) is normally referred to as the in-phase (I) component of the signal and the signal xQ (t) isc 2002 - Michael P. Fitz - The University of California Los Angeles

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6 Complex Baseband Representation of Bandpass Signals

Example 1.1: Consider the bandpass signal

xc(t) = 2 cos(2f m t) 2 cos(2f ct) sin(2f m t) 2sin(2f ct)where f m < f c. A plot of this bandpass signal is seen in Fig. 1.2 with f c = 10f m . Obviously wehave

xI (t) = 2 cos(2f m t) xQ (t) = sin(2 f m t)

andxz (t) = 2 cos(2f m t) + j sin(2f m t).

The amplitude and phase can be computed as

xA (t) = 1 + 3 cos2(2f m t) xP (t) = tan 1 [sin(2f m t), 2cos(2f m t)] .A plot of the amplitude and phase of this signal is seen in Fig. 1.3.

The next item to consider is methods to translate between a bandpass signal and a complexenvelope signal. Basically a bandpass signal is generated from its I and Q components in astraightforward fashion corresponding to (1.1). Likewise a complex envelope signal is generatedfrom the bandpass signal with a similar architecture. Using the results

xc(t) 2 cos(2f ct) = xI (t) + xI (t) cos(4f ct) xQ (t) sin(4f ct)xc(t) 2 sin(2f ct) = xQ (t) + xQ (t) cos(4f ct) + xI (t) sin(4f ct) (1.5)

Fig. 1.4 shows these transformations where the lowpass lters remove the 2 f c terms in (1.5).Note in the Fig. 1.4 the boxes with / 2 are phase shifters (i.e., cos ( / 2) = sin( )) typicallyimplemented with delay elements. The structure in Fig. 1.4 is fundamental to the study of allmodulation techniques.

1.3 Spectral Characteristics of the Complex Envelope

1.3.1 BasicsIt is of interest to derive the spectral representation of the complex baseband signal, xz (t), andcompare it to the spectral representation of the bandpass signal, xc(t). Assuming xz (t) is anenergy signal, the Fourier transform of xz (t) is given by

X z (f ) = X I (f ) + jX Q (f ) (1.6)

where X I (f ) and X Q (f ) are the Fourier transform of xI (t) and xQ (t), respectively, and the energyspectrum is given by

Gx z (f ) = Gx I (f ) + Gx Q (f ) + 2 X I (f )X Q (f ) (1.7)

where Gx I (f ) and Gx Q (f ) are the energy spectrum of xI (t) and xQ (t), respectively. The signalsxI (t) and xQ (t) are lowpass signals with a one-sided bandwidth of less than BT so consequently

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-3

-2

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0

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2

3

0 0.5 1 1.5 2

x c ( t )

Normalized time, f mt

0

0.5

1

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Amplitude Phase

0 0.5 1 1.5 2

A m p l i t u d e

P h

a s e , d e gr e

e s

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1.3 Spectral Characteristics of the Complex Envelope 7

Figure 1.2: Plot of the bandpass signal for Example 1.1.

Figure 1.3: Plot of the amplitude and phase for Example 1.1.

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LPF

LPF

+

-

Complex aseband to andpass Conversion

andpass to Complexaseband Conversion

q t I ( ) q t I ( )

q t Q ( ) ( q t Q

q t c ( )

2 2

2 2cos f t c( ) 2 2cos f t c( )

8 Complex Baseband Representation of Bandpass Signals

Figure 1.4: Schemes for converting between complex baseband and bandpass representations.Note that the LPF simply removes the double frequency term associated with the down conver-sion.

X z (f ) and Gx z (f ) can only take nonzero values for |f | < B T .Example 1.2: Consider the case when xI (t) is set to be the message signal from Example 1.0(com-puter voice saying bingo) and xQ (t) = cos (2000t ). X I (f ) will be a lowpass spectrum with abandwidth of 2500Hz while X Q (f ) will have two impulses located at 1000Hz. Fig. 1.5 show themeasured complex envelope energy spectrum for these lowpass signals. The complex envelopeenergy spectrum has a relation to the voice spectrum and the sinusoidal spectrum exactly aspredicted in (1.6).

Eq. (1.6) gives a simple way to transform between the lowpass signal spectrums to thecomplex envelope spectrum. A similar simple formula exists for the opposite transformation.Note that xI (t) and xQ (t) are both real signals so that X I (f ) and X Q (f ) are Hermitian symmetricfunctions of frequency and it is straightforward to show

X z (f ) = X I (f ) + jX Q (f )X z (f ) = X I (f ) jX Q (f ). (1.8)

This leads directly to

X I (f ) = X z (f ) + X z (f )

2

X Q (f ) = X z (f ) X z (f )

j 2 . (1.9)

Since xz (t) is a complex signal, in general, the energy spectrum, Gx z (f ), has none of the usualproperties of real signal spectra (i.e., spectral magnitude is even and the spectral phase is odd).

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-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000-60

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20Energy spectrum of the complex envelope

Frequency, Hertz

1.3 Spectral Characteristics of the Complex Envelope 9

Figure 1.5: The complex envelope resulting from xI (t) being a computer generated voice signaland xQ (t) being a sinusoid.

An analogous derivation produces the spectral characteristics of the bandpass signal. Exam-ining (1.1) and using the Modulation Theorem of the Fourier transform, the Fourier transformof the bandpass signal, xc(t), is expressed as

X c(f ) = 1 2X I (f f c) +

1 2X I (f + f c)

1 2 j X Q (f f c)

1 2 j X Q (f + f c) .

This can be rearranged to give

X c(f ) =X I (f f c) + jX Q (f f c) 2 +

X I (f + f c) jX Q (f + f c) 2 (1.10)

Using (1.8) in (1.10) gives

X c(f ) = 1 2X z (f f c) +

1 2X z (f f c). (1.11)

This is a very fundamental result. Equation (1.11) states that the Fourier t