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}

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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