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January/February 2018

Faculty of Engineering

Examination for the Degrees

of

Bachelor of Engineering

Master of Engineering

Master of Science

EENG31400 / EENGM1400

DIGITAL FILTERS AND SPECTRAL ANALYSIS

Q.1 Spectral smearing is a problem encountered when analysing a signal with the Discrete Fourier Transform (DFT) due to windowing of the input signal in the time domain. Figures Q1.1 and Q1.2 show the magnitude of the 32-point DFT X[k], k=0,....31 of discrete time signal x[n] = ?!!!! for the case of two windows w[n]:

(a) Indicate which figure corresponds to which choice of window. Justify you answer. (8 marks)

(b) Which normalised frequency F do the following indices k correspond to:

• k = 32

• k = 6          (2 marks)

(c) Would the shape of the spectrum in Figures Q1.1 and Q1.2 change if the length of the DFT increased? Justify your answer. (4 marks)

Suppose that the discrete time signal x[n] = ?!!!! resulted from sampling the continuous time signal ? ? = ?!!!! with sampling period ? = 10!!??? and no aliasing.

(d) What analogue frequency f does index k = 6 corresponds to in the spectrum of Figures Q1.1 and Q1.2? (2 marks)

(e) If ?! is the Nyquist frequency draw the magnitude of the DTFT of x[n] from 0 to 2π (4 marks)

Q.2 (a) The decimation in time (DIT) fast Fourier transform (FFT) relies on a repeated process of split and merge for computing an N-point (N being a power of 2) DFT with less computational effort than a direct implementation. In Figure Q2 below an N=8 point DIT FFT is shown at the first stage of this process, wherein the input sequence x[n] has been split once and two N=4 point DFTs are computed and merged.

Draw the flowgraph of the final N=8 point DIT FFT. Write the input samples x[n] in the correct order at the input (left side) of your flowgraph and include all ‘twiddle’ factors with the correct exponent for each stage of the resulting DIT FFT flowgraph.

(b) Suppose you are told that an N=32 FFT algorithm has a twiddle factor of ?!"! for one of the butterflies in its fifth (last) stage. Is the FFT a decimation-in-time or decimation-in-frequency algorithm? Explain your answer. (4 marks)

(c) A time domain signal, x(t) = cos(2πft), where f = 4Hz, is sampled at 16Hz for 0.25s. Draw the appropriate decimation in time (DIT) FFT flowgraph, showing all complex coefficients and use this to compute the DFT for this signal. (8 marks)

Q.3 (a) Use a Hamming window to design an 8th order (9 tap) band pass FIR filter with a pass-band between 200Hz-400Hz and a sampling frequency of 1200Hz (12 marks)

(b) What is meant by the term linear phase response? (4 marks)

(c) What is the main advantage of a linear phase response? (2 marks)

(d) What are the conditions on the impulse response of an Nth order FIR filter in order for it to have a linear phase response? (2 marks)