1. Figure Q1 below represents a biquadratic digital filter in state-variable realisation. Figure Q1 (a) State the difference equation which relates w[n] to x[n] and the equation which relates y[n] to...

1. Figure Q1 below represents a biquadratic digital filter in state-variable realisation. Figure Q1 (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 XXXXXXXXXX XXXXXXXXXX - + - = z z z z H z . [8 Marks] (b) Show that the transfer function may also be written as ? ? ? ? ? ? - - ? ? ? ? ? ? - = XXXXXXXXXX ( ) z z z z H z . [2 Marks] (c) Carry out a partial-fraction expansion of H(z) to show that the impulse response of the filter may be expressed as XXXXXXXXXXk k h[k] = XXXXXXXXXX . [6 Marks] (d) Draw a realisation of the filter as a parallel combination of two first-order filters. [4 Marks] (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. [5 Marks] B365X Digital Signal Processing Academic Year XXXXXXXXXXa) The figure below shows a block (i) Outline the function of the following blocks: digitisation, FFT, averaging, cartesian (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 averaging (iv) Explain clearly how the calibration operation works, quoting any appropriate formulas (b)In the context of the DFT leakage; scalloping loss; windowing; coherent gain; sidelobes; fall (c)Consider an N-term finite length data sequence x[n]: n=0,..,N write down the sampled waveform which embodies x[n] and state its f discrete spectrum from this, clearly explaining the implicit periodic extension this involves. Explain how the Discrete Fourier Transform X[k] The figure below shows a block-diagram of the processing side of a transfer Outline the function of the following blocks: digitisation, FFT, averaging, cartesian Explain the two ways in which an incorrect setting of the i/p level can affect the measurement Explain why it makes more sense for the averaging to go before the FFT Explain clearly how the calibration operation works, quoting any appropriate formulas DFT, outline the meaning of the following terms: bin; zero leakage; scalloping loss; windowing; coherent gain; sidelobes; fall-off rate. term finite length data sequence x[n]: n=0,..,N-1. Assuming a 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. Page 3 of 7 diagram of the processing side of a transfer-function analyser. Outline the function of the following blocks: digitisation, FFT, averaging, cartesian-to-polar Explain the two ways in which an incorrect setting of the i/p level can affect the measurement to go before the FFT Explain clearly how the calibration operation works, quoting any appropriate formulas [9 Marks] , outline the meaning of the following terms: bin; zero-padding; spectral off rate. [8 Marks] 1. Assuming a sampling period T, domain spectrum. Derive a discrete spectrum from this, clearly explaining the implicit periodic extension this involves. Explain of the sequence x[n] may be derived from this. [