Signal Processing And Digital Filtering

Signal Processing And Digital Filtering

1. a) It is required to design a digital audio effect processor that processes an

audio signal with a highest frequency of 20 kHz. A suitable sampling

frequency is used, and each sample is quantised using an analogue-to-digital

converter that has a full-scale range of 10 V.

i)? Find the minimum number of bits required to represent each sample if the RMS (root-mean-square) quantisation error must be kept below

0.02 mV.?

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ii) How many megabytes (MB) of disk storage would be required to record three minutes of high quality audio in stereo format from the

digital audio effect processor? State any assumption and show all

calculations. ?

3

b) It is required to design a causal digital filter with the following

specifications:

- DC signals do not pass through the system.

- The filter should have a high-pass response with one real pole and

one real zero.

- The pole is at a distance of 0.5 from the origin in the z-plane.

i)? Find the filter transfer function ( ) and plot the corresponding pole- zero diagram.?

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ii) Find the magnitude response | ( )| of the filter, then normalise it

such that | ( )| = 1. ? 4

iii) Find the impulse response ?( ) of the filter in time domain after normalisation. ?

3

 

 

Question continued.

c) A digital signal generator generates discrete samples 0, 1 , 2 , … , −1 over one period. The transfer function of this generator is given by:

( ) = 0 + 1 − 1 + 2 − 2 + … + − 1 − ( − 1 )

Prove, with all the necessary explanations, equations and diagrams, that to

convert the above generator into a periodic sample generator, the transfer

function would be given as:

0 + 1 − 1 + 2 − 2 + … + − 1 − ( − 1 ) ( ) = 1 − − ?

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Total

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2. a) Consider the following two filters:

Filter 1: ( ) = ( ) + ( − 8 ) + ( − 16 )

Filter 2: ( ) = ( ) − ( − 8 ) + ( − 16 )

i) With explanation, suggest what these filters may represent. ? 1

ii) Derive their magnitude frequency response in its simplest form. Show

all intermediate steps. Plot these responses.? 5

iii) Plot the pole-zero diagrams for both filters. 2

b) You are a DSP engineer in a company working with a team to build an echo

cancellation algorithm. A member of the team suggested to start with the

following two equations:

Echo equation:

( ) = ( ) + ( − ) . . . (1)

Square modulus of the DFT of the echo equation:

| ( ) |2 = | ( ) |2 (1 + 2 cos ( ) + 2) . . . (2)

where ( ) is the direct sound, ( ) is the echo signal, represents the round-trip travel time from the source to a reflecting object, and the

coefficient is a measure of the reflection and propagation losses.?

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Question continued.

Then, he suggested to take the natural logarithm (i.e. the ) of Equation (2) above and carry out further processing steps that could lead to cancellation

of the echo and keeping only the useful signal ( ).

You have been asked to explore the feasibility of this method by providing

full analysis supported by all the necessary details, equations and diagrams.

Comment on whether this method is useful or not, and state its main

drawback.

c) Prove that the following five sequences will produce the same DFT/FFT

output for = 4 (i.e. only four frequency points are required):? 3

d) Consider the following two time sequences:

= [ 0, 1, … , −1]

= [ −1, … , 1, 0] Prove that the DFT of the second sequence can be given in terms of that of

the first sequence as follows:

( ) = − ( − ) , = 0, 1, … , − 1

where is the twiddle factor. ?

2

Total

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3. a) It is required to design a high-pass infinite impulse response (IIR) filter using

the Butterworth approximation and the Bilinear transformation with the

following design specifications:

- Sampling frequency is 128 kHz.

- Maximum attenuation of 0.5 dB in the pass-band at = 48 kHz.

- Minimum attenuation of 40 dB in the stop-band at = 8 kHz.

i)? Using pre-warped analogue frequencies and the given attenuations, calculate the minimum order of the analogue filter. ?

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ii) Using the minimum order of the analogue filter obtained in i) above,

calculate Ω , the 3-dB frequency, in the Butterworth approximation

to have exactly 0.5 dB attenuation at Ω .?

2

iii) Determine the transfer functions of both the analogue and,

subsequently, the digital filter (show all steps). ? 4

b) i) ? Design a 4th order low-pass digital Butterworth filter using the Bilinear transformation with cut-off frequency at 2 kHz and a sampling

frequency of 10 kHz. Hence, determine the digital transfer function

( ) and implement the filter in cascade form. Show all the required

steps.?

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ii) Design a low-pass digital Butterworth filter using the Bilinear transformation with -3 dB gain at one-fourth of the sampling frequency

and a gain of at least -14 dB at one-third of the sampling frequency.

Determine the digital transfer function ( ) and implement the filter

in cascade form. Show all the required steps.?

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Total

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4. a) i) Design a length-5 Finite Impulse Response (FIR) band-stop filter with a symmetric impulse response, with the following specifications:

| (0)| = 1 , | ( /2)| = 0 and | ( )| = 0 .6 . Hence, determine the

impulse response ?[ ] and the transfer function ( ). ?

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ii) Draw a direct form realisation of the designed filter in part i) above

using minimum number of multipliers. ? 2

iii) Prove that the filter designed in part i) above has linear phase response. 3

b) When designing FIR filters, the impulse response is usually modified by a

windowing function.

i) Using a suitable window (other than the rectangular), design a 10th order FIR low-pass filter with cut-off frequency at 5 kHz if the

sampling frequency is 40 kHz. Hence, find all filter coefficients. [Hint:

Stop band ripples should be less than -40 dB]. ?

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ii) Draw a direct form realisation of the filter designed in part i) above

using minimum number of multipliers. ? 3

iii) Using the filter you have designed in part ii) above as a starting point,

design a high-pass filter with the same design specifications. ? 3

Total

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