Preparations for exam

1. s- , z- domains
2. Laplace , z- and DFT properties
3. z-transform calculations for functions.
4. Butterfly diagram
5. FIR, IIR filter design concepts
6. Problems based on methods used to convert analog filter into a digital filter
7. Windowing.

Good Day !

Homework - 13

Q1. Realise the following second order system by using direct form - I
          y[n]  =  y[n-1]  -0.5y[n-3]  +  0.5x[n-1]

Q2. Determine the direct form - II realisation for the following system:
          y[n]  =  -0.1y[n-1]  +  0.72y[n-2]  +0.7x[n]  - 0.252x[n-2]

Q3. Determine the transposed structure for the system given by the difference equation :
         y[n] = 0.5 y[n-1]  - 0.25 y[n-2]  +  x[n]  + x[n-1]

Q4. Realise the FIR system H(z) = ( 1 + 0.5z^-1)(1 + 0.5z^-1 + 0.25 z^-2) in
       Direct form and Cascade  form.

Q5. Let the coefficients of a three stage FIR lattice structure be K1 = 0.1, K2 = 0.2, and K3 = 0.3.
       Find the  coefficients of direct form - I FIR filter and draw its block diagram.

Q6. An FIR filter is given by y[n] = x[n] + (2/5) x[n-1] + (3/4)x[n-2] + (1/3) x[n-3].
       Draw the lattice structure.

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Good Day !

Review

We shall now FOCUS on complete understanding of Digital Filter Design. While we do this, it is advised to review all topics covered till now with stress on :

1. Sampling and Nyquist criterion
2. Difference Equation
3. Z-Transform
4. Impulse response
5. Convolution
6. Discrete Fourier Transform
7. Fast Fourier Transform
8. FIR filter design techniques
9. IIR filter techniques

I will upload a quiz and assignment tonight.

Good Day !

Homework - 12

Look at the two figures given below and explain their significance in your own words.

Figure 1 a to f

Figure 2 a to d

Homework - 11

Dear ALL !

If you find time, do study additional material without worrying too much in case you do not understand something.

You will be doing a GREAT FAVOUR to the class by sharing all your doubts with me.
This will help me in preparing for the next class better.

Collect the following graphic representations before next week :

1. Time-domain and frequency domain representation of a rectangular waveform with a time-period of T.
    (Hint : Fourier Series).

Observe : What happens in the frequency domain as T tends to infinity in the time-domain ?

2. Time-domain and frequency domain representation of a pulse (width/duration = Tp). A pulse can be thought of as a rectangular waveform with a time-period of T = Infinity. (Hint : Fourier Transform.)

3. Time-domain and frequency domain representation of a periodic train of  unit impulses with a frequency of fs( = 1/T). (Hint : See chapter on Sampling). Assume fs is the Nyquist rate.

Now, take a signal which is band-limited (No frequencies above the highest frequency say +W/2.) Note : If W/2 is the highest frequency, Nyquist rate fs will be W.

4. What will be the plot in the frequency domain when the above band-limited signal is sampled at its Nyquist rate fs ? Sketch it.

5.What will be the plot in the frequency domain when the above band-limited signal is sampled at LESS THAN the  Nyquist rate fs ? Sketch it.

6. What will be the plot in the frequency domain when the above band-limited signal is sampled at GREATER THAN Nyquist rate fs ? Sketch it.

You will find all the answers in the  text books.
You are encouraged to discuss this with your colleagues to get  a clear picture of signals and their spectra (frequency domain representation).

I will give you some more hints tonight.

I will post the solutions on Sunday.

Good Day.

DSP_Quiz_01

Holi Greetings to ALL !

I have posted a NEW quiz today.
Please checkout.

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Homework - 10

Find the Discrete Fourier Transform of the following sequences :
Assume N = 4 and the samples taken for a duration T = 2pi
1.  x[k] = {1, 1, 0, 0}.
     Ans :  {2, 1-j, 0, 1+j}

2. x[k] = {1, 2, 1, 2}
    Ans :  {6, 0, -2, 0}

3. x[k] = {6, 0, -2, 0}
    Ans : {4, 8, 4, 8}

4. x[k] = {4, 8, 4, 8}
    Ans : {24, 0, -8, 0}

Good Day

Clarification

Dear ALL !

A doubt was raised in the class yesterday as to why k.n must have a range 0 to N-1.

Here is the answer : While k indicates the discrete sample in the time domain and n indicates the sample in the frequency domain...both k and n vary between 0 and N-1. So k.n need not be between 0 and N-1.

But, when we calculate the Euler function using various values of k and n, you will see that all converge into the same set which is equivalent to taking kn between 0 and N-1.

You can try this by calculating the function for (k,n) = (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2),(1,3),(2,0),(2,1),(2,2),(2,3), (3,0),(3,1),(3,2),(3,3).

All values will finally belong to the set kn = {0,1,2,3}.

Please go through the complete explanation by carefully noting the points.


Note : z-Transform, DFT and FFT each have its own advantages and disadvantages which we shall understand indepth once the mathematics is clear.

I will post again in the evening with details of a new quiz.

Good Day !

Update for Week - 3

I despatched a mail to each one which contains a 21 page word document of DFT_Tutorial.
We will use this in today's class.

Our objective is to be well prepared on the following topics for week-4 without fail.
1. Discrete Time Signal representation.
2. Nyquist rate of sampling.
3. Frequency domain representation of signals using Fourier series/Fourier Transform/z-Transform/DFT techniques as applicable.
4. Knowing how to calculate the above transforms.
5. Applying convolution techniques.
6. Knowing how to obtain impulse response of a system.
7. Knowing how to convert a block diagram representation of a system into an I/O representation and vice-versa.

We will have sufficient material and knowledge to begin quizzing on a serious note.

Good Day !

For the class

Dear ALL !

Please bring the graph sheets and other material like last week.
On Sunday, we will do more on z-Transforms and have an introduction to Discrete Fourier Transform.

Good Day

Homework - 9

Answer the following questions by referring to the text book :

1. Can we connect two LTI systems, with impulse responces h1[n] and h2[n], in series ? What will be the output y[n] in such a case ?
2. Extend the above understanding on how to calculate the overall response when independent LTIs are connected in a complex arrangement of series and parallel connections.
3. What do you understand about Region of Convergence (RoC) ?
4. Explain why the z-Transform is incomplete (without RoC) in defining the z-domain representation of a given time-domain function. Given examples.
5. What are the properties of z-Transform ?
6. Locate the z-Transforms and RoC for the following functions from your text book :
1. Impulse function delta[n]
2. Step function u[n]
3. Increasing function a^n u[n]. Read a^n as a raised to the power of n.
4. n (a^n) u[n]
5. -a^n. u[-n-1]
6. -na^n u[-n-1]
7. Note : The inverse z-Transform can be computed by any one of the following methods :
1. Direct evaluation
2. Expansion into a series of terms in the variables z, and z^-1
3. Partial-fraction expansion and table lookups.

Good Day

Update and Review

Dear ALL !

I hope you are slowly getting into the rhythm of DSP. Did you get hold of the text book ? That is the first requirement. Repeat the exercise that you have done in the class once again. Do the homework and post any queries to plan my next class better.

A few students wanted an easier book to follow. Well, I can suggest Digital Signal Processing by P. Ramesh Babu from SCITECH Publications. This is popular among students doing DSP courses. However, you must go through Proakis which discusses lot of fundamental concepts. The book by Ramesh is an exam pass option.

I am sure you all are experienced enough to know the difference.

Please come to the next class with the ready-reference material I gave as homework yesterday.

No homework today :)

Good Day

Homework - 8

Homework is to be taken seriously on a daily basis to keep up with the material and also move forward faster.

Today, prepare a small ready-reference booklet for yourself that contains the following information

1. Mathematical representation of some Elementary Discrete-Time signals like 

• Unit impulse sequence
• Unit step sequence
• Unit ramp signal
• Exponential signal

1. A table showing Properties of z-Transform (You will find this in the chapter on The z-Transform)
2. A table showing common z-Transform pairs 
3. Convolution formula and Properties of convolution.
4. Any other mathematical formulae such as sum of Arithmetic series, Geometric series etc, Calculus and Trigonometry that you may need during the class.

You will find all these from the text book.

Although I would strongly recommend intensive study of the text book T1 to get a strong grip on fundamental concepts, you may refer to other DSP books with the sole aim of coming back to Proakis as soon as your initial fears are removed.

All our future classes will be activity based.

Good Day

Homework - 7

A. Find  the convolution between

1.  x[n] = {0, 1, 2} , h[n] = { 0, 0, 1, 1}

2.  x[n] = {1, 1, 1, 1, 1, 1, 1} , h[n] = {1, 1, 1, 1, 1, 1, 1, 1, }

B. The impulse response is the output of an LTI system when the impulse input or impulse function or impulse signal is applied over a system at n =0, ie. when an input is the impulse function or delta function , the output y[n] will be retained in the same way as that of the system reponse h[n].

C. Remember the following :

1. A LTI system is causal if and only if its impulse response is zero for negative values of n, otherwise it is called non-causal or anti-causal
2. A LTI system is said to be stable if its impulse response is absolutely summable (ie. the sum of all samples (ignoring the sign) between zero and infinity  result in a sum that lies between zero and infinity)

So, we can know a lot about a system from its impulse response !

Quizzes will be updated from today onwards..keep checking.

Good Day !

Homework - 6

1. Review all the previous home works.
2. Please come to the class with the following

• White sheets (at least 5)
• Pencils
• Eraser
• Graph sheets (2)

3. Please note : The quizzes that will be posted are not going to affect your internal evaluation. They are meant to help you do better in the evaluation. There is NO obligation on you to take the quiz as on me to post it :)

Good Day

Homework - 5

Today's homework is more like recollecting a few important points from signals and systems. In case you do not understand a point, you are advised to go back to the text books or knowledgeable friends. In case you have a problem, relax and come to the next class with an open mind.

Good Day

1. Consider the analog signal x(t) = 3cos(100.pi.t). Determine the minimum sampling rate that would restore the signal normally.
Remember and revise :

• Signals and systems are often more usefully described by their frequency-domain properties than their time-domain properties.
• The more a signal is localised in one domain (time or frequency), the less it is localised in the other domain.
• Convolution and multiplication of functions are dual operations in the time and frequency domain.
• The Fourier transform of a periodic signal consists only of impulses.
• Signal energy is conserved in the Fourier transformation process (Parseval's theorem).
• The Fourier series expresses a periodic signal as a sum of sinusoids at harmonics of the fundamental frequency of the signal.
• The sinusoids used in the Fourier series to represent a signal are all orthogonal to each other.
• If a stable Linear Time-Invariant (LTI) system is excited by a periodic signal, the response  is also periodic signal with the same fundamental frequency.
• If a signal is discrete in one domain, it is periodic in another.
• A shift in one domain corresponds to a multiplication by a complex exponential in another domain.
• If a continuous-time signal is sampled to form a discrete-time signal , the Discrete-time Fourier Transform (DTFT)(will be covered next week) of the dicrete-time signal can be found from the Continuous-time Fourier transform(CTFT) of the continuous-time signal by a change of variable, but the converse is not generally true.
• Every LTI system is completely characterised by its impulse response.
• The response of an LTI system to an arbitrary (any) signal can be found by 'convolving' the signal with the system impulse response.
• The impulse response of a cascade connection of LTI systems is the convolution of the individual impulse responses.
• The impulse response of a parallel connection of LTI systems is the sum of the individual impulse responses.
• A LTI system is BIBO(Bounded input produces bounded output) stable if its impulse response is absolutely summable.
• LTI systems can be represented by block diagrams and this type of representation is useful both in synthesising systems and in understanding their dynamic-behaviour.

Bye

Homework-4

1. When our filmi hero in movies is chasing (in a tonga) the villain (driving a car) why do the tonga wheels appear to be moving backwards ?
2. How can we understand the 'highest frequency' in terms of a visual image , such as a photograph or a TV show ?
3. A discrete-time signal x[n] is defined as
              (1 + n/3) for -3 < n < -1
  x[n] =   1            for 0 < n < 3
               0             elsewhere

a. Determine its values and sketch the signal x[n]
b. Sketch the signals that result if we :
      b1. First fold x[n] and then delay the resulting signal by four samples.
      b2. First delay x[n] by four samples and then delay the signal
c. Sketch the signal x[-n + 4]
d. Compare the results in part b. c. and derive a rule for obtaining the signal x[-n + k] from x[n].


Good Day

T1 Edition -reg

Amit raised a query regarding the edition of the text book T1.Although the syllabus suggests 1998 edition, I am reading  the 2007 edition.You can buy any edition available as long as it is after 1998.Also, some of you might have free downloaded the pdf version from the net ?

The book I am referring to is Digital Signal Processing - Principles, Algorithms and Applications by John G. Proakis, Dimitris G. Manolakis ( I have the 4th edition 2007)

Bye

Homework - 3

1. Find out more about the following functions

• Rectangular pulse
• Sinc pulse or Sine over argument pulse
• Signum function
• Raised Cosine pulse

2. How are the above functions represented in the frequency domain ?

Good Day

Homework-2

1. When we compare two amplifiers, we do this using some important characteristics like Gain, Bandwidth, Input impedance and Output impedance. Similarly, when we compare two A/D converters, what characteristics are most relevant for comparison.
2. A deterministic signal can be represented mathematically. How will you represent a random/probabilistic signal mathematically ie. what 'numbers' and 'parameters' will you associate to a random signal ?
3. What is a Gaussian function ? What is the Fourier transform of a Gaussian function ?

I am designing a quiz on DSP for your use. Please send your individual mail ids. I will send the username and password.

Good Day
DSRao

Homework-1

You will find enough hints from the chapters 1 and 2.
1. What is the Nyquist rate for an electrocardiogram (ECG) signal ?
2. What is SQNR ?
3. Show the Block Diagram Representation of  a "moving average filter"
4. What are the 4 steps in a convolution operation.
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I will be sending individual mails with userid and password so that you can login to my quiz site and practise answering simple questions from each chapter on a daily basis.

Good Day
DSRao

Digital Signal Processing

Dear ALL !

Welcome to my blog on DSP.
I will post necessary guidelines and links that may help in your study goals.
Please do feel free to post your comments and suggestions to make the course worthwhile and fun.

Check this article to get a start :
http://www.redcedar.com/learndsp.htm

Take care
DSRao