Page 80 - Fundamentals of Communications Systems

2.32 Chapter Two

Problem 2.20. An energy spectrum has a peak value of 0.02. Find the value the
energy spectrum must go below to be

(a) 40-dB below the peak.
(b) 3-dB below the peak.
Problem 2.21. A concept that sometimes finds utility in engineering practice is
the concept of RMS bandwidth. The RMS bandwidth is defined as

∞

2
W R = f G x (f )df (2.98)
−∞
For the energy spectrum in Figure 2.14 find the RMS bandwidth.
Problem 2.22. A concept that often finds utility in engineering practice is the
concept of group delay. For a linear system with a transfer function denoted

H(f ) = H A(f ) exp[ jH p (f )] (2.99)

the group delay of this linear system is defined as

1 d
τ g (f ) =− H p (f ) (2.100)
2π df

Group delay is often viewed as the delay experienced by a signal at frequency
f when passing through the linear system H(f ).

(a) An ideal delay element has an impulse response given as h(t) = δ(t − τ d ),
where τ d > 0 is the amount of delay. Find the group delay of an ideal delay
element.
(b) An example low pass filter has

1
H(f ) = (2.101)
j 2π f + a
Plot the group delay of this filter for 0 ≤ f ≤ 5a.
(c) Give a nontrivial H(f ) that has τ g (f ) = 0. In your example is the filter
causal or anticausal?
Problem 2.23. (RW) An interesting characteristic of the Fourier transform is lin-
earity and linearity can be used to compute the Fourier transform of complicated
functions by decomposing these function into sums of simple functions.

(a) Prove if y(t) = x 1 (t) + x 2 (t), then Y (f ) = X 1 (f ) + X 2 (f ).
(b) Find the Fourier transform of the signal given in Figure 2.15.