Design the digital filter using the bilinear transformation

Question

1. Figure Q1 below represents a biquadratic digital filter in state-variable realisation.

(a) State the difference equation which relates w[n] to x[n] and the equation which relates y[n] to w[n]. From these, show that the z-domain transfer-function of the filter is given by

H(z)= (0.15z -0.25z²) / (z² - 0.6z +0.08)

(b) Show that the transfer function may also be written as H(z) = {(0.25z/(z-0.2))((0.6-z)/(z-0.4))}

(c) Carry out a partial-fraction expansion of H(z) to show that the impulse response of the filter may be expressed as

h[k] = 0.25 (0.4^k)-0.5 (0.2^k)

(d) Draw a realisation of the filter as a parallel combination of two first-order filters.

(e) Starting from the transfer function, H(z), derive the difference equation which specifies the current output y[n] as a suitable combination of previous outputs and current and previous inputs.

2. (a) The figure below shows a block-diagram of the processing side of a transferdiagram of the processing side of a transfer-function analyser.

(i) Outline the function of the following blocks:  digitisation, FFT, averaging, cartesian-to-polar

(ii )Explain the two ways in which an incorrect setting of the i/p level can affect the measurement

(iii) Explain why it makes more sense for the averagingExplain why it makes more sense for the averaging to go before the FFT

(iv) Explain clearly how the calibration operation works, quoting any appropriate formulas,

b) In the context of the DFTDFT, outline the meaning of the following terms: bin; , outline the meaning of the following terms: bin; zerozero-padding; spectral leakage; scalloping loss; windowing; coherent gain;leakage; scalloping loss; windowing; coherent gain; sidelobes; fall sidelobes; fall-off rate.off rate.

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

3. (a) Signals may be categorised as either continuous or discrete. They may also be categorised as either periodic or aperiodic. These categorisations are equally meaningful in the time domain or in the frequency domain. Draw a copy of the table below in your answer book
and fill it in, entering one of the following codes in each box:

• CA (continuous aperiodic)
• CP (continuous periodic)
• DA (discrete aperiodic)
• DP (discrete periodic).

Each code should appear twice in the table.

Time Domain	Frequency Doma	Connection
		ourier transform
		Sampled Signal
		DFT
		Fourier Series

(b) You are required to design a low-pass digital filter with a sampling frequency of 48kHz and a cut-off frequency at 8kHz. The theoretical pulse response of an ideal low-pass filter which exactly meets this specification is

h[k] = 1/3sinc(k/3)

(i) Explain why it is necessary to truncate and delay this response in order to have a filter that can be implemented in practice.

(ii) Outline the consequences of the truncation on the amplitude response of the filter. Outline the consequences of the delay on the amplitude response of the filter.

(iii) State the effects on the amplitude response of applying a window to the truncated coefficients

(iv) Outline the potential impact on the amplitude response of representing the filter coefficients by finite-precision binary numbers

(v) If the filter is to be implemented using a DSP chip which is capable of carrying out one MAC every 20ns, explain why the implementation can only allow a maximum of 1041 MACs per output sample.

(c) A band-limited signal whose maximum frequency fmax is less than some value fN may be expressed as a frequency-domain Fourier series:

Show by inverse Fourier transformation that its waveform is given by

4. (a) State the three categories of processing we wish to do to signals and state three advantages of doing them digitally.

(b) You are required to design a digital filter, to operate at a sampling frequency of 48 kHz, which has a phase shift of -π/2 radians at 8 kHz. The transfer function of the analogue prototype you are using is H(s)= (α-s)(α+s) . This is an all-pass filter whose magnitude response is unity at all frequencies and the frequency at which it has a phase shift of -π/2 radians is α ω = α.

(i) Design the digital filter using the bilinear transformation and show that the transfer function of the resulting filter may be approximately expressed as:

H(z) = (1.5774 - 0.4226z)(1.5774z - 0.4226)

Calculate its z-plane poles and zeros.


(ii) Calculate the phase shift of the filter at 12kHz and at 24kHz

(c) By considering points on the z-plane of the form z = e^j?  , 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(?/2)

Show how the whole of the ω -axis, from -∞ = ω to +∞ = ω , is covered by this transformation.