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Assignment 2 and Submission Guidelines

School School of Information Technology and Engineering

Course Name Master of Engineering (Telecommunications)

Unit Code ME606

Unit Title Digital Signal Processing

Assessment Author Dr. Reza Berangi

Assessment Type Assignment 2 (Individual)

Assessment Title z-Transforms, Filters Concepts

Unit Learning Outcomes covered in this assessment

a. Develop and implement signal processing algorithms in Matlab b. Undertake in-depth design of digital filters

Weight 15%

Total Marks 100

Word/page limit N/A

Release Date Week 7

Due Date Week 11 (31 May 2019, 11:55 pm)

Submission Guidelines

• All work must be submitted on Moodle by the due date along with a completed Assignment Cover Page.

• The assignment must be in MS Word format, single line spacing, 11-pt Calibri (Body) font and 2 cm margins on all four sides of your page with appropriate section headings.

• Reference sources must be cited in the text of the report, and listed appropriately at the end in a reference list using IEEE referencing style for School of Business and School of Information Technology and Engineering respectively.

Extension If an extension of time to submit work is required, a Special Consideration Application must be submitted directly through the AMS. You must submit this application three working days prior to the due date of the assignment. Further information is available at: http://www.mit.edu.au/about-mit/institute-publications/policies-proceduresand-guidelines/specialconsiderationdeferment

Academic Misconduct

Academic Misconduct is a serious offence. Depending on the seriousness of the case, penalties can vary from a written warning or zero marks to exclusion from the course or rescinding the degree. Students should make themselves familiar with the full policy and procedure available at: http://www.mit.edu.au/aboutmit/institute-publications/policies-procedures-and-guidelines/PlagiarismAcademic-Misconduct-Policy-Procedure. For further information, please refer to the Academic Integrity Section in your Unit Description.

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© MIT/SITE | ME606 Digital Signal Processing Assignment 2 of 9

ME606 Assessment 2

Introduction

The objective of this assessment is for each student to demonstrate understanding of the

contents of lecture materials in the unit by applying the principles and concepts in lecture

notes. Concepts covered by this assignment include z-transforms, and FIR filter design.

Section 1. FIR filter design

In general, we have 3 commonly used FIR filter design techniques

(1) Windowed Fourier series approach

(2) Frequency sampling approach

(3) Computer-based optimization methods

In this assignment we practice the first and the second methods in designing FIR filters

1-1 Designing a low pass FIR filter using Windowed Fourier Series approach

The amplitude frequency response of an ideal low pass filter is shown in Figure. 1. Its

impulse response can be found from its inverse Fourier transform as:

h = (ω𝑐

π) ∗ sinc (

nω𝑐

π) , n =. . , −2, −1,0,1,2, … , 0 < ω𝑐<π (1)

Figure 1. Ideal low pass filter amplitude frequency response (left) and impulse response (right)

Using the equation (1) we can find all the impulse response samples. The filter is IIR but by

truncating the impulse response to a limited number of samples we can make it an FIR filter.

To do so we usually select M samples (M is usually odd number) around n=0 as shown in

Figure 2.


Shift by M/2

Figure 2. Truncated impulse response

By multiplying the resultant impulse response by a window we can reduce unwanted ripples

in the spectrum.

The following Matlab code designs a FIR filter with Windowed Fourier Series approach.

wp=pi/8; % lowpass filter bandwidth M=121; n=-(M-1)/2:(M-1)/2; % selection time window h0=(wp/pi)*sinc((wp/pi)*n); % truncated impulse response h=h0.*rectwin(M)'; % windowing % h=h0.*hamming(M)';% windowing % h=h0.*hanning(M)';% windowing % h=h0.*bartlett(M)';% windowing % h=h0.*blackman(M)';% windowing figure;plot(h)

ylabel('Impulse response') xlabel('Samples')

% spectrum FFTsize=512; pxx=20*log10(abs(fft(h,FFTsize))); fxx=(0:(FFTsize/2)-1)*(pi/(FFTsize/2)); figure;plot(fxx,pxx(1:FFTsize/2));

ylabel('Amplitude[dB]');

xlabel('frequency [Radian]');grid on

Do the following tasks and plot all the graphs in your report: 1) Run the above program and plot the impulse response and the amplitude spectrum

of the filter.

2) Measure the filter frequency response at p=pi/8. Is it close to the expected value of -6dB? How much is the maximum ripple in the pass band.

3) This filtering technique does not define any cutoff. If we assume the stopband starts at -20dB, measure the ratio of the transient band to the pass band.

4) Increase the filter impulse response length to M=255 and run the program. a) If we assume the stopband starts at -20dB, measure the ratio of the transient

band to the passband. b) Measure the ripple in the passbad.


c) Discuss the effect of the filter length on the ripples and stopband attenuation. 5) Use Hamming and Hanning window with the filter impulse response length, M=121,

and discuss the effect of the window on the ripples and stopband attenuation.

6) Change p=pi/4 and repeat 1) and discuss the effect.

1-2 Designing a bandpass FIR filter using Windowed Fourier Series approach

A FIR lowpass filter can be converted to a bandpass filter by multiplying its impulse response

by a complex sinusoidal with the frequency of the center frequency of the desired bandpass

filter. In fact, the whole filter band shifts by this frequency. The following MATLAB code

designs such a bandpass filter:

wp=pi/16; % lowpass filter bandwidth M=121; n=-(M-1)/2:(M-1)/2; % selection time window h0=(wp/pi)*sinc((wp/pi)*n); % truncated impulse response h1=h0.*rectwin(M)'; % windowing w0=pi/4; % BPF center frequency h=h1.*exp(j*w0*n); % Lowpass to bandpass conversion

figure; subplot(211);plot(real(h));ylabel('Real part of impulse response') xlabel('samples') subplot(212);plot(imag(h));ylabel('imag part of impulse response') xlabel('samples') % spectrum FFTsize=512; pxx=20*log10(abs(fft(h,FFTsize))); fxx=(0:(FFTsize/2)-1)*(pi/(FFTsize/2)); figure;plot(fxx,pxx(1:FFTsize/2));ylabel('Amplitude

[dB]');xlabel('frequency [Radian]');grid on

Do the following tasks and plot all the graphs in your report:

7) Run the above program and plot the impulse response and the amplitude spectrum of the filter.

8) What are the passband edges p1 p2. Measure the attenuation in frequency response at these frequencies and compare them with the expected value of -6dB.

9) Measure the maximum ripple in the passband.

10) Increase p to pi/4 and discuss the result 1-3 Designing a band-stop FIR filter using Frequency sampling approach

In this section, you must write your own code to design a band stop filter using the

knowledge you gained from the sections 1-1 and 1-2


Write a code to design a band-stop filter. Design a band-stop filter by having the

following block diagram in your mind

The spectrum mask is as the following:

11) Report your MATLAB code. Select the filter impulse response length M=121;

12) Plot the frequency response, |H(ej)|, of your filter on the top of the spectrum mask in a

single plot.

13) Plot 20*log10(|H(ej)|) of your filter on the top of the spectrum mask in dB in a single

plot.

14) Measure the amplitude at =[/4, 3/4, /2] on your second plot.

15) Find the deepest point in your frequency response in dB. How you can increase the

depth of the stopband?

1-4 Designing a low pass FIR filter using Frequency sampling approach

In the frequency sampling approach, we design an amplitude response in the frequency

domain and find the impulse response by applying IFFT on that frequency response.

The following MATLAB code designs a low pass FIR filter using Frequency sampling

approach. It initially designs the amplitude spectrum and the apply IFFT. Using some shifts it

finds the truncated impulse response.

wp=pi/8; M=121; FFTsize=512; % passband frequency samples Np=fix((wp/(2*pi))*FFTsize); % number of pass band samples Ns=FFTsize/2-Np; % number of stop band samples H1=[ones(1,Np) zeros(1,Ns+1)]; H=[H1 H1(end:-1:2)]; % sampled frequency spectrum; fxx=(0:(FFTsize/2)-1)*(pi/(FFTsize/2)); figure;plot(fxx,H(1:FFTsize/2)); ylabel('Amplitude'); xlabel('frequency [Radian]');grid on

BPF

+ Band-stop

filter X[n] y[n] -

+

0

1

|H(ej)|


% finding impulse response by truncating h1=real(ifft(H)); h0=[ h1(end-(M-1)/2+1:end) h1(1:1+(M-1)/2)]; figure;plot(h0) ylabel('Impulse response') xlabel('Samples')

% spectrum pxx=20*log10(abs(fft(h0,FFTsize))); fh=figure;plot(fxx,pxx(1:FFTsize/2));hold on;

% effect of windowing

h=h0.*hanning(M)';% windowing

% spectrum pxx=20*log10(abs(fft(h,FFTsize))); figure(fh);plot(fxx,pxx(1:FFTsize/2));grid on;

legend('rectwin','hanning')

Do the following tasks and plot all the graphs in your report:

16) Run the above program and plot the impulse response and the amplitude spectrum of the filter. Try to understand what the program does.

17) Use the above program as your base, design a band pass filter that passes the frequencies between pi/8<w<pi/4. Only modify the lines 5-7 and do not touch the rest of the program.

Section 2. Analyzing a system in Z and time domain

A digital system is given by the following transfer function:

𝐻(𝑧) =1 − 0.2𝑧−1

(1 − 0.5𝑧−1)(1 − 0.3𝑧−1)

(1)

18) Assume the system is causal. Draw

its region of convergence (ROC) in

the z-plane

19) Is the system stable or unstable?

Why?


20) Determine the system difference

equation in the form of

knxbknyaM

k

k

N

k

k 00

(2) 21) Give the values for ai and bj , N and M

22) Sketch the magnitude of the frequency response of the system between 0 and

23) Suppose the above transfer function is a filter, what type of filter is it? (a low-pass,

high-pass, band-pass, band-stop or all-pass)

24) Sketch a block diagram for implementing this system

25) Find the system causal impulse response (use partial fraction decomposition of the

equation (1) )

26) Assume the initial condition is zero, enter a unit impulse to the system and find the

impulse response of the system h[n] for the first 10 samples and compare it with the

step11

Section 3. Filtering

Use the filter impulse response “h” you found by running the given program in the section 1-3 with a Hanning window.

Run the following code

x=randn(1,5000); % signal to be filtered NFFT=512; % FFTT size (N) overlap samples=0; % number of overlap samples in time frames y=conv(x,h); % filtering operation using convolution [Pxx,W] = pwelch(y,hamming(NFFT),overlap_samples,NFFT); figure;plot(W,Pxx);xlabel('W [radian] from 0-pi');ylable(' power spectral

density')

27) Plot and include the plot in your report. Simply, mention the application of the

“pwelch” program, and what are its input parameters.

Run the following Matlab code

x=[-zeros(1,500) ones(1,1000) zeros(1,500)]; % signal to be filtered y=conv(x,h); % filtering operation using convolution

figure;plot(x,'.-') hold on;plot(y,'.-'); grid;

xlabel('sample') legend('filter input signal','filter output')


28) Measure the filter delay and find a ratio of the filter delay to the filter impulse response length

Format of the Report Your report should be typed using a suitable equation editor. Hand written reports will not be

accepted.

Marking criteria

Section included in the report

Marks

1 17

2 9

3 2

Total 28

Marking Rubric for Exercise Answers

Grade Mark HD 80%+

D 70%-79%

CR 60%-69%

P 50%-59%

Fail < 50%

Excellent Very Good Good Satisfactory Unsatisfactory

Evaluation

Logic is clear and easy to follow with strong arguments

Consistency logical and convincing

Mostly consistent and convincing

Adequate cohesion and conviction

Argument is confused and disjointed

Sophistication and effectivity

The presented solution demonstrated an extreme degree of sophistication

The presented solution demonstrated a high degree of sophistication

The presented solution demonstrated an average degree of sophistication and effectivity to secure

The presented solution demonstrated a low degree of sophistication and effectivity to secure

The presented

solution demonstrated a poor degree of sophistication and effectivity to secure

Explanation All elements are present and well integrated.

Components present with good cohesion

Components present and mostly well integrated

Most components present

Lacks structure.

Reference style Clear styles with excellent source of references.

Clear referencing/ style Generally good referencing/style

Unclear referencing/style

Lacks consistency with many errors


Report structure and report presentation

Proper writing. Professionally presented

Properly written, with some minor deficiencies

Mostly good, but some structure or

Acceptable presentation Poor structure, careless presentation

presentation problems

The End

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