Problems – Chapter 1
1.1 Determine the energy spectral density of a square pulse x(t) = rect(t/T). Calculate
the normalized energy Ex in the pulse.
1.2 Find the average normalized power in the waveform x(t) = 10cos10t + 20cos20t.
1.3 Determine which, if any, of the following functions have the properties of
autocorrelation functions. Why?
1 for − 1 ≤ τ ≤ 1
-4
x 10
1
(a ) x (τ) = 0.9
0 otherwise
0.8
0.7
(b) x (τ) = δ(τ) + sin 2πf 0 ( τ) 0.6
0.5
(c) x (τ) = exp(| τ |) 0.4
0.3
1− | τ | for − 1 ≤ τ ≤ 1 0.2
(d) x (τ) = 0.1
0 otherwise 0
-10 -8 -6 -4 -2 0 2 4 6 8 10
Figure 1
1.4 Determine which, if any, of the following functions have the properties of power
spectral density functions. Why?
(a ) X(f ) = δ(f ) + cos 2 2πf
(b) X(f ) = 10 + δ(f − 10)
(c) X(f ) = exp( −2π | f − 10 |)
(d) X(f ) = exp[ −2π(f 2 − 10)]
1.5 The Fourier transform of a signal x(t) is defined by X(f) = sinc f. Find the
autocorrelation functions of signal x(t).
2
− 4 sin[ π(f − 10 )10 ]
6 −4
1.6 Given the PSD: S(f ) = 10 . Find the value of signal
π(f − 106 )10 −4
bandwidth using the following bandwidth definitions:
(a) Half-power bandwidth
(b) Null-to-null bandwidth
(c) 99% of power bandwidth
(d) Bandwidth beyond which the attenuation is 35dB
(e) Absolute bandwidth.
1.7 The two-sided PSD S(f) = 10-6f 2, of a waveform x(t) is shown in Figure 1.
(a) Find the normalized average power in x(t) over the frequency band from 0 to 10
kHz
(b) Find the normalized average power contained in the frequency band from 5 to 6
kHz
Questions – Chapter 1
1.8 Describe the differences between analog and digital signals
1.9 Identify three reasons why digital communication systems are better than analog
communications systems
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