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Note for Digital Signal Processing - DSP By Lokesh Lodha

  • Digital Signal Processing - DSP
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Jaipur National University, Jaipur School of Engineering & Technology Department of Electronics & Communication Sub: DSP&A Assignment-1 Q1. A Continuous time periodic signal with fourier series cofficients Ck = (½) |k| a period Tp = 0.1 sec. passes through an ideal ideal low pass filter with cutoff frequency Fc = 102.5 Hz. The resulting signal ya (t) is sampled periodically with T = 0.005 sec. Determine the spectrum of the sequence y(n) = ya (nT). Q2. Consider the sampling of the bandpass signal whose spectrum is illustrated in figure given below. Also determine the minimum sampling rate Fs to avoid aliasing. Q3. A bandlimited continuous-time signal xa (t) is sampled at a sampling frequency Fs ≥ 2B. Determine the energy Ed of the resulting discrete-time signal x(n) as a function of the energy of the analog signal, Ea , and the sampling period T = 1/Fs. Q4. Given the eight-point DFT of the sequence 1, 0 ≤ 𝑛 ≤ 3 x (n) = { } 0, 4 ≤ 𝑛 ≤ 7 compute the DFT of the sequences1, 𝑛=0 (a) X1 = { 0, 1 ≤ 𝑛 ≤ 4 } 1, 5 ≤ 𝑛 ≤ 7 0, (b) X1 = {1, 0, 0≤𝑛≤1 2 ≤ 𝑛 ≤ 5} 6≤𝑛≤7 Q5. Compute the 8-point DFT of the sequence 1, 0 ≤ 𝑛 ≤ 7 x (n) = { } 0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 by using the decimation-in-frequency FFT algorithm described in the text. Q6. Computed the 16-point DFT of the sequence X(n) = cos (π/2) n , 0≤ n ≤ 15

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Using the radix-4 decimation-in-time algorithm. Q7. Determine the order of the analog Butterworth filter that has a (-2) db pass band attenuation at a frequency of 20 rad/sec and atleast (-10) db stop band attenuation at 30 rad/sec. Q8. Consider a Chebychev filter with attenuation not more than 1.0 dB for | w |1000 rads / s and at least 10 dB for | w |5000 rads / s. Q9. Let us design a Butterworth LP filter with specifications: Passband amplitude to be within 2.0 dB for frequencies below 0.2π rad / s and the stopband magnitude to be less than 10 dB for 0. 4π rad / s or above. Assume unity gain at D.C. Q10. A non-recursive filter is characterized by the transfer function H(z) =1 + 2z + 3z2+ 4 z3 + 3 z4+ 2 z5+ z6 z6 Find the group delay. ******************

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