EL3147 Digital Signal & Image Processing Semester 2 2018-19

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EL3147

Digital Signal & Image Processing (A)

Semester 2

2018-19

Q1. A touch-tone phone dialler uses Dual-Tone Multi-Frequency (DTMF) signals to encode keys pressed, and pressing key ‘3’ generates a signal containing the sum of the two tones at 697 Hz and 1477 Hz. Figure Q1 shows a signal processing system to detect pressing of this key. It consists of an anti-aliasing filter with a frequency response of H(f), an analogue-to-digital (A/D) converter with a conversion speed set to 0.125 ms, a 256-point fast Fourier transform (FFT), and a peak detector which indicates pressing of key ‘3’ when the magnitude values at 697 Hz and 1477 Hz produced by FFT exceed a threshold value.

Figure Q1

(a) If the anti-aliasing filter before A/D is removed, identify a pair of the lowest aliasing frequencies which could be combined to trigger a false indication of key ‘3’ being pressed.          (3 marks)

(b) If the anti-aliasing filter before A/D is a second order Butterworth filter with the magnitude frequency response given by

where fs denotes the sampling frequency, determine the required increase in the signal magnitude for the aliasing frequency obtained in (a) to produce the false presence of 1477 Hz at the FFT output.   (7 marks)

(c) If the A/D conversion time is halved, determine whether the false presence of 1477 Hz at the FFT output is more likely or less likely with respect to the result obtained from (b).  (6 marks)

(d) Suggest two possible solutions which may be used to minimise the probability of false indication.  (4 marks)

TOTAL for Q1: 20 marks

Q2. A biological signal is acquired using a sampling frequency of 1 kHz and is smoothed using a 4-tap moving average filter to reduce local signal fluctuation. Since this is a symmetrical and even order FIR (finite impulse response) filter, its frequency response is given by

(a) If the input signal to the moving average filter consists of two components at 70 Hz and 250 Hz with equal amplitude value at 100 mV, determine the corresponding output amplitude values.  (10 marks)

(b) If the spectral properties of the smoothed signal are to be analysed by using the discrete Fourier transform (DFT) with a frequency resolution of 5 Hz, determine an appropriate record length required.  (2 marks)

(c) Determine the frequency index corresponding to 70 Hz by applying the DFT to the smoothed signal based on the record length obtained from (b). (2 marks)

(d) Explain two possible errors contained in the spectrum for accurate frequency analysis of the biological signal, and suggest possible solutions to reduce their effects in the frequency spectrum.  (6 marks)

TOTAL for Q2: 20 marks

Q3. A bandpass filter can be implemented by cascading in series a lowpass filter and a highpass filter. Working in a team to develop a bandpass filtering system for an audio recording studio, your role is to implement the digital filter for the highpass part and you are given the following filter specifications:

Sampling frequency             16 kHz

Passband edge frequency     4.2 kHz

Stopband edge frequency     3.8 kHz

Passband ripple                   0.03

Stopband attenuation           0.02

(a) Using Table Q3, determine an appropriate window function and an appropriate order for the design of a FIR (finite impulse response) highpass filter to satisfy the above filter specifications.  (4 marks)

(b) Applying 8-point sampling of the ideal highpass frequency response at regular intervals along the frequency axis upto the sampling frequency, give a set of appropriate frequency magnitude values for the design of digital filter.  (4 marks)

(c) Based on the window function obtained from (a) and the set of sampled frequency magnitude values obtained from (b), determine the linear difference equation for 5-tap FIR filter by using the inverse discrete Fourier transform (IDFT) given by

(12 marks)

Table Q3

TOTAL for Q3: 20 marks

Q4 (a) Explain and contrast the following aspects of IIR and FIR filters:

(i) linear difference equation  (2 marks)

(ii) impulse response  (2 marks)

(iii) pole positions  (2 marks)

(iv) stability  (2 marks)

(b) A filter H(z) has two zeros and two poles, as defined below:

Zeros:   z1 = -0.4

z2 = 0.4

Poles:   p1 = 0.8 + j 0.2

p2 = 0.8 - j 0.2

The d.c. magnitude response of the filter is known to be 10, and the filter is implemented with a sampling frequency of 120 kHz.

(i) Determine a mathematical expression representing the transfer function H(z) of the digital filter.

HINT: remember to include the scale factor K.  (4 marks)

(ii) Show your working, determine the magnitude response of the filter at a frequency of 30 kHz.  (4 marks)

(iii) Sketch a Direct Form II flow-graph for the filter, label all signal flow directions, delay elements, and filter coefficients.  (4 marks)

TOTAL for Q4: 20 marks

Q5. Convolution is getting increasingly important in digital signal & image processing. It can be applied to smooth the data when noise suppression is needed. It can also be used to detect features such as image/signal edges. With the surging demand of smart devices, convolution techniques has found lots of interesting applications in developing learning systems such as computer aided diagnosis.

Figure Q5. The raw image (left) and the blue channel of the 200th row (right).

Figure Q5 above shows an image and the blue channel’s intensity of the 200th row. High values (~230) in Figure Q5(b) indicate white background, while small values (50~100) correspond to the object pixels.

Table Q5 shows two small segments from Figure Q5(b).

Table Q5. Two segments from 150th row of coins.png.

(a) Suppose we have a smoothing kernel g1: = [0.25, 0.5, 0.25]. Apply this kernel g1, via convolution, to the segment 1 in Table Q5. Denote the output, i.e., smoothed segment 1, as y1.

(i) Write the equation of y(i), where i is the index of y, by convoluting kernel g1 with segment 1. (2 marks)

(ii) Show the numerical result of (i) above. Use 0s for padding. (3 marks)

(b) Apply an edge extraction kernel g2 := [-0.5, 0, 0.5], via convolution, to the segment 2 in Table Q5. Denote the output as y2.

Show numerical results of the convolution between kernel g2 and segment 2. Use 0s for padding. (3 marks)

(c) If f(x) has n1 elements and k(x) has n2 elements, typically how many non-zero elements there might be in the full convolution of f(x) and k(x)? (2 marks)

(d) For 2D convolution, suppose the image is 50x50 and the kernel is a 7x7 Gaussian kernel.

(i) What is the image size of full convolution operation between the image and the kernel? (2 marks)

(ii) What is the image size of valid convolution operation between the image and the kernel? (2 marks)

(e) Suppose a convolution neural network (CNN) is designed to conduct object recognition. Assume that the input image of CNN is 23x23 and the output is 3x3. Design the architecture of the CNN, including the sizes of the convolution kernels. (6 marks)

TOTAL for Q5: 20 marks

Q6 Histogram equalisation is a very popular algorithm in image enhancement due to its simplicity. Figure Q6 below shows an image (a) and its result of histogram equalisation (b). Figure Q6(c) is a small portion of the data from Figure Q6(a).

Figure Q6. Image histogram equalisation: (a) the original image, (b) output of histogram equalisation, (c) intensity data for the 5x5 area of interest in (a).

(a) (i) Construct a histogram with the data in Figure Q6(c). Assume the bins are centred at {5, 15, 25, …, 245, 255}. You can use Table Q6 below to help you. (Note: do NOT forget to put the results into answer book) (3 marks)

(ii) Use the histogram constructed in (i) above to estimate the mean intensity of the image data. Show details of your calculations. (2 marks)

(b) Describe the steps for applying histogram equalisation. (5 marks)

(c) When applying histogram equalisation, the following equation is used:

where gin and gout are the image intensities of the input/output of histogram equalisation. p(gin) and p(gout) are the normalized histogram values of the input/output of histogram equalisation at gin and gout respectively.

(i) In histogram equalisation, we assume that p(gout) is a constant. Explain why this assumption is used in histogram equalisation. (3 marks)

(ii) Use the equation above, derive the equation used in calculating gout with a given gin. (7 marks)

Table Q6. Histogram and its equalisation

TOTAL FOR Q6: 20 marks