Problems_ Chapter 1

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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