Page 67 - Fundamentals of Communications Systems

Signals and Systems Review 2.19

Definition 2.15 A time-invariant system is one in which a time shift in the input only
changes the output by a time shift, i.e.,

x(t − τ) → y(t − τ) (2.72)

A linear time-invariant (LTI) system is described completely by an impulse
response, h(t). The output of the linear system is the convolution of the input
signal with the impulse response, i.e.,

∞

y(t) = x(λ)h(t − λ)dλ = x(t) ∗ h(t) (2.73)
−∞
The Fourier transform (Laplace transform) of the impulse response is denoted
the frequency response (transfer function), H(f ) = F{h(t)}(H(s) = L{h(t)}),
and by the convolution theorem for energy signals we have

Y (f ) = H(f )X(f ) (Y (s) = H(s)X(s)) (2.74)

Likewise the linear system output energy spectrum has a simple form

2
∗
G y (f ) = H(f )H (f )G x (f ) =|H(f )| G x (f ) = G h (f )G x (f ) (2.75)

EXAMPLE 2.29
If a linear time-invariant filter had an impulse response given in Example 2.2, i.e.,

sin(2πWt)
h(t) = 2W = 2Wsinc(2Wt)
2πWt
it would result in an ideal lowpass filter. A filter of this form has two problems for an
implementation: (1) the filter is anticausal and (2) the filter has an infinite impulse
response. The obvious solution to having an infinite duration impulse response is to
truncate the filter impulse response to a finite time duration. The way to make an
anticausal filter causal is simply to time shift the filter response. Figure 2.6 (a) shows
the resulting impulse response when the ideal lowpass filter had a bandwidth of W =
2.5 kHz and the truncation of the impulse response is 93 ms and the time shift of
46.5 ms. The truncation will change the filter transfer function slightly while the delay
will only add a phase shift. The resulting magnitude for the filter transfer function
is shown in Figure 2.6 (b). If the voice signal of Example 2.3 is passed through this filter,
then due to Eq. (2.75), the output energy spectrum should be filtered heavily outside of
2.5 kHz and roughly the same within 2.5 kHz, as shown in Figure 2.7. Figure 2.7 shows
the measured energy spectrum of the input and output and these measured values match
exactly that predicted by theory. This is an example of how communications engineers
might use linear system theory to predict system performance. This signal will be used
as a message signal throughout the remainder of this text to illustrate the ideas of
analog communication.