Digital Signal Processing

(a) Explain DFT. Give matrix relations for computing DFT and IDFT.

(b) Why FFT is so important? What are its advantages? Draw the complete flow diagram of DIT FFT algorithm, taking sequence length N=8.

(c) Derive the frequency response of a linear phase FIR filter with anti- symmetric impulse response and filter length N even.

(d) Define the Chebyshev polynomial C_N(x) Obtain the recursive relation to build up higher order Chebyshev polynomials.

(e) What are the advantages of poly-phase decomposition?

(a) Show that the Up-sampler and the Down-Sampler are Linear but Time-varying system. 

 (b) What do you understand by Zero input limit cycle oscillations. Explain with the help of an example.

(a) Prove the following property of DFT when X(k) is the N-point DFT of sequence x(n).

(i) X(k) is real and even when x(n) is real and even

(ii) X(k) is imaginary and odd when x(n) is real and odd.

(b) Derive the relation for Bilinear Transformation connecting s-domain and the z-domain. Show the mapping of points in s-domain to z-domain.

(c) Derive the frequency response of a linear phase FIR filter with symmetric impulse response and filter length N odd.

(d) Draw the Direct form-II and its transposed structure for the given transfer function. 

(e) Show that the Up-sampler and the Down-Sampler are Linear but Time-varying system. 

(a) For a N-point periodic sequence x(n) with DFT X(k), prove that

(b) Define circular convolution. Evaluate circular convolution of the following sequences.     x(n) = {1,3,4,2,1} and h(n) = {2,0,1,0, 1}. 

(b) Define circular convolution. Evaluate circular convolution of the following sequences.     x(n) = {1,3,4,2,1} and h(n) = {2,0,1,0, 1}. 

(a) Find inverse DFT of 

(b) Calculate the IDFT using Decimation-in-Time FFT structure for the given coefficient. 

(b) Calculate the IDFT using Decimation-in-Time FFT structure for the given coefficient. 

(a) Convert the analog filter the system functioninto a digital IIR filter using Impulse Invariant technique. Assume T=1 sec.

(b) Design a FIR filter with the following desired frequency response,  

Using Hamming window for N = 7. 

(b) Design a FIR filter with the following desired frequency response,  

Using Hamming window for N = 7. 

(a) Using Bilinear Transformation, design a Butterworth HPF to meet the following specifications: 

(b) Write the various window functions used for FIR filter design.Compare their important performance parameter. 

(a) Discuss and explain Hilbert transform.

(b) Explain the Levinson Durbin Algorithm in detail.

(a) Draw and explain the Lattice realization of an all-pole filter function.

(b) Without factoring any polynomial, determine whether or not the following filter function is stable. 

 (a) Prove the identity.

(b) Develop an expression for the output y(n) as a function of input for a given multi-rate system.  

(a) A cascaded realization of the two-first order system is given by H(z) = H₁ (z). H₂(z). Find out the overall output noise power of the system. Where 

 (b) What do you understand by Zero input limit cycle oscillations. Explain with the help of an example.

(a) Prove the following property of DFT when X(k) is the N-point DFT of sequence x(n) -

(i)  X(k) is real and even when x(n) is real and even.

(ii) X(k) is imaginary and odd when x(n) is real and odd. 

(b) Prove the circular convolution property of DFT.

(a) If X(k) is an N-point DFT of x(n) and if x(n)=-x{N-1-n). Then show that X(0) = 0.

(b) Calculate the IDFT using Decimation-in-Frequency FFT structure for the given coefficient.

X(k) - {38,-5.828 + j6.07, j6,-0.172 +j8.07,10,-0.172 - j8.07,-j6.-5.828 - j6.07} 

fa! A requirement exists for a low pass FIR filter satisfying the following specifications: 

Pass band                             0-5 KHz

Sampling Frequency            18 KHz

Filter Length                           9

Obtain the filter coefficients using Frequency Sampling method. 

(b) Write the various window functions used for FIR filter design. Compare their important performance parameter.

Derive the relation for Impulse-Invariant Technique connecting s-domain and the z-domain. Show the mapping of points in s-domain to z-domain.

Using Bilinear Transformation, design a Butter worth LPF to meet the following specifications:

(a) Draw the Direct form - and its transposed structure for the given transfer function. 

Draw and explain the Ladder-Lattice realization of the following transfer function. 

a) Explain the Lavinson Durbin Algorithm in detail.

(b) Without factoring any polynomial, determine whether or not the following filter function is stable.

(b) Express the output y(n) of Figure (a) as a function of input x(n). By simplifying the expression derived show that y(n) = x(n-1)

(b) Express the output y(n) of Figure (a) as a function of input x(n). By simplifying the expression derived show that y(n) = x(n-1)

(a) What do you understand by Zero input limit cycle oscillations, Explain with the help of an example.

(b) Prove that

What are the disadvantages of Analog Signal Processing and the advantages with Digital Signal Processing? 

Determine the convolution of  with any single x(t).

Determine the z transform of 

Determine discrete time Fourier Transform of - discrete time signal x(n)=rⁿ, un) for |r|<1. 

Explain FFT Discrete in time and Discrete in Frequency FFT. 

Draw basic element of Signal Processing.

Prove Frequency Shifting and Frequency differentiation property of Fourier transform. 

Draw block diagram representation in the direct form cascade form and parallel form for a discrete time L T I system represented by the following transfer function. 

Find the Discrete time ID FT of Y(K)= {1,0, 1, 0}.

Using FFT and IDFT determine the output of system iſ input x(n) and impulse response h(n) are given as under:

x(n)= {2,2,4}

H(n)= {1,1}

Define and explain linear phase system.

Obtain the structure of cascade and parallel realization of the following transfer function:

Briefly explain Goertzel Algorithm for efficient DFT computation.

Explain in place computation.

Compute the 8 point circular convolution of the sequences.

Use the rectangular Window to design a linear FIR filter of order N-24 to approximate for the following frequency response magnitude. 

Differentiate between FIR filter and IIR filters.

What is the condition for the impulse response constant group and phase delay?

A digital low pass IIR filter is to be designed with Butterworth approximation using bilinear Transformation technique. Find the order of filter to meet the following specifications.

i) Pass band magnitude is constant with 1dB for frequencies below

ii) Stop band attenuation in greater than 15dB for frequencies between 

Design the digital high pas filter for cut off frequency of 30H_z and sampling frequency of 150 Hz using BLT. 

Design the second order low puss digital filter of the Butterworth type using BLT for the specification given below:

i) Analog transfer function of the filter 

Design the second order low puss digital filter of the Butterworth type using BLT for the specification given below:

ii) Cut off frequency=1 KHz

Design the second order low puss digital filter of the Butterworth type using BLT for the specification given below:

iii) Sampling frequency-10 KHz

A digital low pass IIR filter is to be designed with Butterworth approximation using bilinear Transformation technique. Find the order of filter to meet the following specifications. 

i) Pass band magnitude is constant with 1dB for frequencies below 0.2

A digital low pass IIR filter is to be designed with Butterworth approximation using bilinear Transformation technique. Find the order of filter to meet the following specifications. 

stop band attenuation in greater than 15aB for frequencies between 0.3 to  .