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EENG31400 DIGITAL FILTERS AND SPECTRAL ANALYSIS
创建日期：2024-08-14 17:02:55
所属分类：数学mathematics
标签： EENG31400
DIGITAL FILTERS AND SPECTRAL ANALYSIS

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DIGITAL FILTERS AND SPECTRAL ANALYSIS

Q.1 Figure Q1 shows the spectrum X[k] resulting from applying a DFT of length L=2048 to discrete time signal x[n] = cos(2πn /14) + 0.75cos(4πn /15) using a Kaiser window w[n]. Figures Q1(a), Q1(b) and Q1(c) show what the spectrum looks like when the length M of window w[n] is equal to Ma, Mb and Mc respectively. Answer the following questions.

(a) Which normalised angular frequency ? does index k=1024 correspond to?      (2 marks)

(b) Which normalised frequency F does index k=1024 correspond to?          (2 marks)

(c) The dashed lines in Figure Q1 indicate the DFT indices k0 and k1 that correspond to the nearest DFT frequencies of the cosine components in x[n]. What is the value of k0 and k1?          (3 marks)

(d) Which of the three window lengths (Ma,Mb,Mc) is the largest and which one is the smallest? Justify your answer.         (5 marks)

(e) Assume that the discrete time signal x[n] resulted from sampling continuous time signal x(t) with a sampling frequency fs=210Hz and no aliasing. What is signal x(t) equal to?      (3 marks)

Q.2 Figure Q2.1(a) shows the spectrum (DTFT) X(?) of a discrete time signal x[n]. Figure Q2.1(b) shows the spectrum of the discrete time signal x[n] in Q2.1(a) after a sampling rate conversion (a change in the sampling rate).

(a) By looking at the spectrum of Figure Q2.1(b), state what sampling rate conversion has taken place (up-sampling or down-sampling) and by what factor.     (4 marks)

(b) Consider the system of Figure Q2.2 for changing the sampling rate of x[n] by a non-integer factor. Determine the output xd[n] if x[n] = cos(3πn/ 4), L=6 and M=7.

(c) Suppose that a continuous time speech signal sc(t) with the Fourier transform. spectrum Sc(ω) shown in Figure Q2.3(a) is processed by the system shown in Figure Q2.3(b) to produce the discrete time signal sr[n]. H(?) in Figure Q2.3(b) is an ideal discrete-time low pass filter with a cut-off frequency of ?c and a gain of L in the passband. The signal sr[n] will be used as input to a speech encoder that operates correctly only on discrete-time speech signals sampled at an 8-kHz rate. Choose values of L, M and ?c that produce the correct input signal sr[n] for the speech encoder.     (6 marks)

Q.3 (a) Following the windowing method of FIR filter design, use a Hamming window to design a 3-tap FIR lowpass filter with a cut-off frequency of 1000 Hz and a sampling rate of 10 kHz.       (10 marks)

(b) Compute the magnitude frequency response of the filter for frequencies Ω = 0, 4/π, 2/π, 4/3π, π. (5 marks)

Q.4 Consider the following digital filter with transfer function:

(a) Sketch the pole-zero diagram.          (4 marks)

(b) Sketch the magnitude of the frequency response in the range 0 ≤ Ω < 2π.    (4 marks)

(c) Draw the 2-stage (cascade) Direct Form. II (canonic form) architecture for this filter.       (3 marks)

(d) What would change in the frequency response of the filter if the denominator of the transfer function became equal to (1-0.9z-1 + 0.81z-2 )(1+ 0.81z-2)

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