1. The pulse response of a digital system obeys the recursion: h[k ] -0.6h[k -1] +0.08h[k -2] {0.5,-0.16,0,0,L}

(a) Show that the transfer function of the system is given by: H(z)= 0.5z(z -0.32)/(z -0.2)(z -0.4)

(b) Draw a realisation of the system as a cascade of two first-order systems.

(c) If the input sequence to the system is x[n] and its output sequence is y[n], derive the difference equation which specifies the current output in terms of previous outputs and current and previous inputs.

(d) Carry out a partial fraction expansion of H(z) to show that the closed form expression for the pulse response is:

h[k] 0.3×(0.2^k) +0.2 ×(0.4)^k.

(e) Draw a realisation of the system as two first-order systems in parallel.

(f) Explain why it is generally good practice to implement a high order system as a cascade or parallel combination of low order systems.

2.  (a)  The diagrram below shows the five stages of processinng in an FFT-based speectrum analyser. 

Outline thhe function(ss) of each bloock.

(b) You have been supplied with a DSSP chip capable of carrying out one rreal MAC eveery 5ns and you are requiredd to use it tto implemennt an FFT-baased audio spectrum annalyser capable oof a 1 Hz resolution overr the full auddio range froom 20 Hz too 20 kHz. Asssuming audio is sa mming the input mmpled at 44.1 kHz and tthe spectrumm analyser iss using a Ha window wwhose 6-dB width is 1.81 bins, calculate the length in samples of the data segment needed to achieve this resolutionn and the mminimum sizze of the FFT this
requires. Determine tthe resultantt sample spacing along thhe frequencyy axis and froom this the numbber of frequuency sample points too be used too cover thee full audio range.

Calculate the numberr of real MAACs required in each stagge of the annalyser (otheer than digitisatioon) and from this deducee the maximuum refresh raate of the annalyser's dispplay.

(c) The continuous Hammming windoww is given by::

Find its sppectrum W(f)

3. (a) An FIR digital filter is required for an oversampled audio DAC. The audio was originally sampled at 44.1 kHz, the DAC is using 16-times oversampling and the processor implementing the filter takes 20ns to carry out one multiply-and-add operation.

(i) Explain, with the aid of suitable diagrams, what oversampling is in this context and  what the benefit(s) of doing it are.

(ii) Calculate the number of multiply-and-add operations the processor can use to workout the numerical value of each output sample of the digital filter and from this deduce the maximum filter order that can be implemented. Explain your logic clearly.

(iii) The theoretical pulse response of an ideal low pass filter is:

h[n] = 1/16 sinc (n/16)

State the reasons why this response cannot be implemented as it stands and explain how the windowing method enables a practical filter to be derived from it. Outline the impact of the windowing on the pass-band, transition band and stop-band properties of the filter.

(iv) Briefly explain how an FIR filter could be designed using an alternative approach based on the DFT.

(b) Consider an N-term finite length data sequence x[n]: n=0,..,N-1. Assuming a sampling period T, write down the sampled waveform which embodies x[n] and state its f-domain spectrum. Derive a discrete spectrum from this, clearly explaining the implicit periodic extension this involves. Explain how the discrete Fourier Transform X[k] of the sequence x[n] may be derived from this.

4. (a) The transfer function of a general digital filter may be written in the following two ways:

On the basis of the second of these, state the pulse response h[k] for this filter and use it to explain clearly why this type of filter is commonly known as an infinite impulse response filter.

b) A digital notch filter is required to remove an unwanted sine-wave from audio digitised at 48kHz. It is proposed to base the design on an analogue prototype filter whose transfer function is H(s) = {s² + ω₀²}/{(s - α)² + ω₀²}

Assuming ω₀² = 1.42517088 × 10¹⁰ and α= -10⁴, find the s-plane poles and zeros of H(s).

(ii) Show that the notch frequency of the prototype filter is 19kHz and calculate the dB-gain of the prototype at DC.

(iii) If you were using a standard bilinear transformation method, calculate the value of ω₀²  which would need to be used in the prototype formula to yield a digital filter with a notch at 19kHz. Explain clearly why this is not the same value as the ω₀²  assumed in (i) above.

(c) By considering points on the z-plane of the form z = e^jΩ, show clearly how the bilinear transformation maps the unit circle on the z-plane onto the ω-axis of the s- plane according to the formula ω= (2/T)tan (Ω/2). Show how the whole of the ω-axis,
from -∞ = ω to +∞ = ω , is covered by this transformation.