complex base band representation

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ECE 461 Fall 2006

August 31, 2006Complex Baseband Representation

Channel Model for Point-to-Point Communications

Point-to-point communications systems are well modeled using a bandpass additive noise channelmodel of the form shown in Figure 1.

s(t) h(t)y(t)

n(t)

r(t)fc fc f

|S(f)|

Figure 1: Real bandpass channel model for point-to-point communications

The message bearing signal s(t) is a real-valued bandpass signal whose spectrum is concentratedin the vicinity of some carrier frequency fc.

Distortions introduced by the channel are characterized by a linear time invariant system withimpulse response h(t), and frequency response H(f) concentrated around fc. The channel re-sponse h(t) may or may not be known at the receiver. In the simplest case, the response h(t)corresponds to a an ideal bandpass filter with bandwidth corresponding to that of the signal s(t).

The additive noise process n(t) is usually idealized by White Gaussian Noise (WGN). The received signal r(t) is a real-valued bandpass process as well.

In the following we convert the bandpass channel model into a more convenient and equivalentcomplex baseband channel model.

Complex baseband representation for signal

Since the signal s(t) is real, its spectrum S(f) is symmetric about f = 0. Hence all of the informationabout the signal s(t) is contained in the positive half of the spectrum S(f), which we define to be

S+(f) = 2S(f)11{f0} . (1)

where u() is the unit step function. The factor of2 in the above equation makes the signal s+(t)have the same energy as the signal s(t). The inverse Fourier transform of the spectrum S+(f) iseasily shown to be the complex signal

s+(t) =1

2[s(t) + j s(t)] , (2)

cV.V. Veeravalli, 2006 1


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where the signal s(t) is the Hilbert transform of s(t) (i.e., the Fourier transform of s(t) is jsgn(f)S(f)).The signal s+(t) is called the pre-envelope of s(t). If we shift the spectrum of S+(f) down to theorigin, we get the baseband signal s(t) with

S(f) = S+(f + fc), and s(t) = s+(t)ej2fct . (3)

Note that since S(f) is not necessarily symmetric around the origin, the signal s(t) is in generalcomplex-valued. The signal s(t) is called the complex envelope or the complex baseband represen-tation of the real signal s(t). From (2) and (3), we get

s(t) = Re[

2 s+(t)] = Re[

2 s(t)ej2fct] . (4)

The complex envelope s(t) can be written in terms of its real and imaginary parts as

s(t) = sI(t) + jsQ(t) . (5)

From this and (4) we get

s(t) =

2[sI(t)cos2fct sQ(t)sin2fct] =

2 a(t)cos[2fct + (t)] , (6)

where

a(t) =

s2I(t) + s2Q(t), and (t) = tan

1 sQ(t)

sI(t). (7)

The signal a(t) is called the envelope of s(t), and (t) is called the phase of s(t). It is to be notedthat every bandpass signal can be written in the forms given in (6).

Equation (6) also suggests a practical way to generate the (components of) complex envelope fromthe passband signal. It is easy to see that if we multiply s(t) by

2cos(2fct) and low-pass filter

(LPF) the output, we produce sI(t). Similarly, if we multiply by

2sin(2fct) and LPF theoutput, we get sQ(t).

The conversion from passband to baseband and vice-versa is illustrated below in Figure 2

s(t) s(t)

sI(t)

sQ(t)

LPF

LPF

2cos2fct

2sin2fct

2cos2fct

2sin2fct

Figure 2: Conversion from passband to baseband and vice-versa.

Complex baseband representation of channel response

Referring to Figure 1, since the output of the channel y(t) is a bandpass signal, it has the complexbaseband representation y(t) = y+(t) e

j2fct. The signal y(t) is related to s(t) through the convo-lution integral, i.e., y(t) = h s(t). The question that we ask now is whether the complex envelopes



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s(t) and y(t) are related in a similar fashion, and if so, what is the corresponding complex impulseresponse? You will show in HW#2 that this in indeed the case and that the corresponding complexbaseband channel response is given by:

H(f) =1

2H+(f + fc) =

h(t) =

1

2h+(t)e

j2fct and h(t) = 2Re[h(t)ej2fct] . (8)

Note the additional factor of

2 in the equation relating h(t) and h(t).

Note thaty(t) = h s(t) = (hI + jhQ) (sI + jsQ)(t) (9)

implies that the I and Q components of y(t) can be computed separately as

yI(t) = hI sI(t) hQ sQ(t) , and yQ(t) = hI sQ(t) + hQ sI(t) . (10)

This suggests a way to implement the passband filter h using real baseband operations.


complex base band representation

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