Chapter 1 : Introduction · Web viewWhat is the relationship between Z transform and the Discrete...

CS2403-DIGITAL SIGNAL PROCESSING DEPARTMENT OF CSE

DEPARTMENT OF ECE

SUBJECT: DIGITAL SIGNAL PROCESSING CODE:CS 2403SEM.: VII DEPT.:CSE

PART - A (Q&A)

UNIT I SIGNALS AND SYSTEMS

Basic elements of DSP – concepts of frequency in Analog and Digital Signals – sampling theorem – Discrete – time signals, systems – Analysis of discrete time LTI systems – Z transform – Convolution (linear and circular) – Correlation.

UNIT II FREQUENCY TRANSFORMATIONS

Introduction to DFT – Properties of DFT – Filtering methods based on DFT – FFT Algorithms Decimation – in – time Algorithms, Decimation – in – frequency Algorithms – Use of FFT in Linear Filtering – DCT.

UNIT III IIR FILTER DESIGN

Structures of IIR – Analog filter design – Discrete time IIR filter from analog filter – IIR filter design by Impulse Invariance, Bilinear transformation, Approximation of derivatives – (HPF, BPF, BRF) filter design using frequency translation

UNIT IV FIR FILTER DESIGN

Structures of FIR – Linear phase FIR filter – Filter design using windowing techniques, Frequency sampling techniques – Finite word length effects in digital Filters

UNIT V APPLICATIONS

Multirate signal processing – Speech compression – Adaptive filter – Musical sound processing – Image enhancement.

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PART A (Q&A)

UNIT I SIGNALS AND SYSTEMS

Q. 1. What is DSP?

Ans. DSP is defined as changing or analysing information which discrete sequences of numbers.

Q. 2. What are the limitations of digital signal processing?

Ans. The digital signal processing systems have many advantages. Even though there are certain disadvantages as follows

1. Bandwidth limitations : In case of DSP, if input signal is having wide bandwidth then it demands for high speed ADC. This is because to avoid aliasing effect, the sampling rate should be atleast twice the bandwidth. Thus such signals require fast digital signal processors. But always there is a practical limitation in the speed of processors and ADC.

2. System complexity : The digital signal processing system makes use of converters like ADC and DAC. This increases the system complexity compared to analog systems. Similarly in many applications the time required for this conversion is more.

3. Power Consumption: A typical digital signal processing chip contains more than 4 lakh transistors. Thus power dissipation is more in caps systems compared to analog systems.

4. DSP systems are expensive as compared to analog system.

Q. 3. What are the applications of DSP.

Ans. The applications of DSP are given below

1. Image processing like pattern recognition, animation, robotic vision, image enhancement.

2. Instrumentation and control like spectral analysis, noise reduction, data compression.

3. Speech/Audio like speech recognition, speech synthesis, equalisation.

4. Biomedical like scanners ECG analysis, patient monitoring

5. Telecommunication like in echo cancellation, spread spectrum and data communication.

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6. Military like Sonar processing, radar processing, secure communication.

7. Consumer applications like digital audio and video, power like monitor.

8. Automotive applications like vibration analysis, voice commands and cellular telephones.

9. Industrial applications like rabotics and CNC, power line monitors.

Q. 4. Define terms ‘signal’ and ‘system’?

Ans. A ‘signal’ may be defined as a physical quantity which varies with time, space or any independent variable Example — voltage, current A ‘system may be defined as a combination of devices and networks or subsystem chosen to do a desired action Example Electrical N/W, mechanical system

Q 5 Write the major classification of signals’

Ans. There are various types of signals Every signal is having its own characteristic The processing of signal mainly depends on the characteristics of that particular signal So classification of signal is necessary Broadly the signal are classified as follows

1 Continuous and discrete time signals

2. Continuous valued and discrete valued signals.

3. Periodic and non periodic signals.

4 Even and odd signals

5. Energy and power signals:

6 Deterministic and random signals

7. Multichannel and multidimensional signals.

Q. 6. Explain sampling function of sinc function.

Ans In mathematics, the sinc, function, denoted by sinc(x) and sometimes as Sa (x),

has two definitions, In digital processing and information theory, the normalized sinc function is commonly defined by

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In mathemetics, the historical unnormalized sinc function is defined by

In both cases, the value of the function at the removable singularity at zero, usually calculated by l’Hospital rule, is something specified explicitly as the limit value 1 The sinc function is analytic everywhere.

Q. 7. What are energy and power signals?

Ans. The energy E of a signal x(n) is defined as

The energy of a signal can be finite or infinite. If E is finite then x(n)

is called an energy signal.

Many signals that posses infinite energy, have a finite average power. The average

power of a discrete time signal x(n) is defined as

If E is finite, P = 0. On the other hand, If E is infinite, the average power may be

either finite or infinite. If P is finite (and non zero), the signal is called a power signal.

Q. 8. Differentiate between linear-Nonlinear system.

Ans. A system is called linear, if superposition principle applies to that system. This

means that linear system may be defined as one whose response to the sum of the

weighted inputs is same as the sum of the weighted responses.

Let us consider two systems defined as follows.

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Here x1(t) is the input or excitation and y1(t) is its output or response and

Here x2 (t) is the input or excitation and y2(t) is its output or response

Then for a linear system

Where a1 and a2 are constants.

Linearity property for both continuous time and discrete time systems may be written

as for continuous time system

For discrete time system

For any non-linear system, the principle of super-position does not hold true and

equations (3) and (4) are not satisfied.

Few examples of linear system are filters, communication channels etc.

Q. 9. Define periodic and non periodic signals Give an example in each case.

Ans A periodic signal repeats after fixed period But non-periodic signal never repeats

Periodic signal like x(t) sin wt and Non periodic signal like A discrete time signal is periodic, if its frequency can be expressed as a ratio of two integers i.e.

Here k and N are integer and N is the period of discrete time signal

Q 10. What is scaling of discrete time signals?

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Ans Scaling of discrete time signals is divided into two parts

Time Scaling Operations

As the name indicates, time scaling operations are related to the change in time scale. There are two types of time scaling operations.

• Down scaling (Compression)

Up scaling (Expansion)

(n) Amplitude Scaling Operation

As the name indicates, in case of amplitude scaling operations, amplitude of signal is changed Different amplitude scaling operations are as follows

• Upscaling (Amplification)

• Downscahng (Attenuation)

• Addition

• Multiplication

Q 11 What is the difference between static and dynamic discrete time signals?

Ans. There can be static and dynamic discrete time systems but cannot be signals

Q 12. Define a discrete time unit sequence functions

Ans. A discrete time unit signal is denoted by U(n) Its value is unity for all positive values of n. That means its value is one for n 0. While for other values of n, its value is zero.

In form of sequence it can be written as

Graphically it is represented as shown below

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Q. 13. Define a discrete time unit ramp function.

Ans. A discrete time unit ramp function is denoted as Ur (n) and it is defined as

Figure below shows the graphical representation of a discrete unit ramp function.

Q. 14. Define transfer function of a system.

Ans. A system may be defined as a set of elements or functional blocks which are connected together and produces an output in response to an input signal. The response of the system depends upon transfer function of the system.

Mathematically it is defined by

Where x(t) is input or excitation

y(t) is 0/P or response

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h(t) is transfer function of the system.

Q. 15. State the necessary and sufficient condition for stability of LTI systems

Ans. LTI system is stable if its impulse response is absolutely summable i e

Here h(k)= h(n) is the impulse response of LTI system Thus equation (1) give the

condition of stability in terms of impulse response of the system.

Now the stability factor is denoted by ‘s’.

Q. 16. What are the constraints on the transfer function if it were to represent a causal LTI system?

Ans. If h(n) is the response of released LTI system to a unit impulse applied at n = 0, it follows that h(n) = 0 for n < 0 is both a necessary and a sufficient condition for causality Hence on LTI system is causal if and only if its impulse response is zero for negative values of n.

Q 17. Define LTI system’

Ans. If a system has both the linearity and time in varience properties, then the system is called as linear time m varient (LTI) system

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Q 18. What are the conditions for the region of convergence of a causal LTI system?

Ans. A discrete time LTI system is causal if and only if the ROC of its transfer function is the extension of a circle, in including infinite

A discrete time LTI systems which has a rational transfer function H(z) will be causal if and only if.

(z) The ROC is the extension of a circle outside the outermost pole and

(ii) Units H(z) expressed as a ratio of polynomials in z, the order of the numerator should be smaller than order of denomenator.

Q. 19. State sampling theorem.

Ans. A continuous time signal x(t) can be completely respresented in its sampled form and recoverd back from the sample form if the sampling frequency

where ‘W’ is the maximum frequency of the continuous time signal x(t)

Q. 20. Convolve {1,3,1) and (1,2,2,).

Ans.

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y (n) is output of the convolution.

Q. 21. What is the difference between stable astable system?

Ans.

Q. 22. Differentiate time variant from time invariant system.

Ans. A system is called time invariant if its input output characteristics do not charge

with time. A LTI discrete time system satisfies boths the linearity and the time invariance properties.

To test if any given systems is time invariant, first apply an arbitrary sequence x (n) and find y (n).

y (n) = T [x (n)]

Now delay the input sequence by k samples and find output sequence denote it as. y(n,k) T[x(n-k)]

Delay the output sequence by k samples denote it as

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For all possible values of k, the systems is the invariant on the other hand

Even for one value of k, the system is time variant.

the output.

Even for one value of k, the system is time variant.

Q. 23. What are symmetric and asymmetric signals?

Ans. An even signal is that type of signal which exhibits symmetry in the time domain This type of signal is identical about the origin Mathematically, an even signal must satisfy the following condition.

For a continuous-time signal, x (t) = x (— t)

For a discrete-time signal, x (n) x (— n)

Figure shows continuous-time and discrete-time even signals.

Similarly, an odd signal is that type of signal which exhibits anti-symmetry. This type of signal is not identical about the origin Actually, the signal is identical to its negative Mathematically, an odd signal must satisfy the following condition

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For a continuous-time signal, x (t) = x (- t)

For a discrete-time signal, x (n) — x (— n)

Figure shows continuous-time and discrete-time odd signals.

Q. 24. What is the frequency response of a discrete LTI system? Derive the frequency response of a system whose impulse response is given by h(n) = a” u(n —1) for (a) <1.

Ans. The frequency response of a linear time invariant discrete time system can be obtained by applying a spectrum of the input sinusoids to the system. The frequency response gives the gain and phase response of the system to the input sinusoids at all frequencies. Let us consider, the inpulse response of an LTI discrete time system is h(n) and the input x(n) to the system is complex exponential e1u. The output of the system y(n) can be

Given

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Q. 25. Determine the power and energy of the unit step sequence.

Ans. The average power of the unit step signal is

Consequently, the unit step sequence is a power signal. Its energy is infinite.

Q. 26. Sketch the block diagram representation of the discrete time system described by the input-output relation.

where x(n) is the input and y (n) is the output of the system.

Ans. According to the question, the output y(n) is obtained by multiplying the input x(n) by 0.5, multiplying the previous input x(n -1) by 0.5, adding the two products and then adding the previous output y(n -1) multiplied by 1/4. Figure below illustrates this block diagram realization of the system. A simple rearrangement of equation is

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Q. 27. Determine if the system y(n) = x(— n) is time variant or time invariant.

Ans. The system is described the input output relation

The response of the system to x (n — k) is

Now if we delay the output y (n), by k units in time, the result will be

Since the system is time variant.

Q. 28. Determine if the system described by the following input-output equation are linear or nonlinear i.e.

Ans. For two input sequences x1(n) and x2(n), the corresponding outputs are

The output of the system to a linear combination of x1(n) and x2(n) is

Finally a linear combination of the two outputs

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By comparing (i) with (ii), we can conclude that the system is linear.

Q. 29. Determine the system y(n) = x (2n) is causal or non causal system.

Ans. y(n) = x(2n)

For output, need of advance value of I/P, so y(n) is non-causal system.

Q. 30. Consider the special case of a finite duration sequence given as:

Resolve the sequence x(n) into a sum of weighted impulse sequences.

Ans. Sequence x(n) is non-zero for the time instants n = - 1,0,2, we need three impulses

at delay k = — 1, 0 ,2

Q. 31. Consider a system with impulse response Determine whether the system is stable or unstable.

Ans.

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Q. 32. Differentiate between a causal and non causal system. Show

that

Ans. A system is causal if the response or output does not begin before the input function is applied. This means that if input is applied at t=t0 then for.

causal system, output will depend on values of input

A causal system is non-anticipatory. The response or output of the causal system to an input does not depend on future values of that input, but depends only on the present or past values of the input.

Causal systems are physically realizable. On the other hand if the response of the system to an input depends on the future values of that input, then the system is non- causal or anticipatory. Non-causal systems cannot be implemented practically. This means that there is no system possible practically which can produce its output before input is

applied.

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Q. 33. Define z-transform of a discrete time signal.

Ans. The Z-transform of a discrete time signal x(n) is defined as the power series

where z is a complex variable. The Z-transform of a signal x(n) is denoted by

whereas the relationship between x(n) and X(z) is indicated by

The z-transform is a infinite power series, it exists only for those values of z for

which this series converges. The region of convergence (ROC) of X(z) is the set of all s values ofz for which X(z) attains a finite value. Thus any time we cite a z-transform. We should also indicate its ROC.

Q. 34. What is Region of convergence?

Ans. The z-transform is an infinite power series, it exists only for those values of z

for which the series converges. The region of convergence (ROC) of X (z) is set of all values of z for which X (z) attains a finite value. The ROC of a finite duration signal is

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the entire z-plane, except possibly the point . These points are excluded because z-n (when n > 0) becomes unbounded for z = and zn (when n > 0) becomes unbounded for z = 0.

Q. 35. What is the relationship between Z transform and the Discrete Fourier transform?

Ans. Let us consider a sequence x(n) having z-transforrn with ROC that includes the

unit circle. If X(z) is sampled at the N equally spaced points on the unit circle. If X(z) is

sampled at N equally spaced pomts on the unit circle.

We obtain

Expression is (2) identical to the Fourier transform X(w) evaluated at the N. equally spaced. Frequencies

If the sequence x(n) has a finite duration of length N or less, the sequence can be

recovered from its N-point DFT. Hence its Z-transform is uniquely determined by its N-point DFI’. Consequently, X(z) can be expressed as a function of the DFT {X(k)} as

follows

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When evaluated on the unit circle (3) yields the Fourier transform of the finite duration sequence in terms of its DFT in the form:

This expression for Fourier transform is a polynomial interpolation formula for X(w)

expressed in terms of the, values {x(k)) of the polynomial at a set of equally spaced

discrete frequencies

Q. 36. What are the application’s of z-transform?

Ans. 1. z-transform is an important tool in the analysis of signals and linear time invarient systems.

2. It is used for the analysis of discrete time systems in frequency domain which in generally more efficient than time domain analysis.

3. It is used for filtering process.

4. Causality of discrete time LTL system.

5. Stability of discrete time LTI system.

6. Determination of poles and zeros of rational z-transform.

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Q. 37. What are the conditions for the region of convergence of a non causal LTI system.

Ans. The condition for non-causal of discrete time LTI system is that the impulse response of a causal discrete time LTI system js given as

This means that h (n) is two sided.

Also, transfer function H(z) is the z-transform of h

(n). The ROC of H (z) of non-causal discrete time LTI

system is the entire z-plane except

Q. 38. State and prove convolution property of z transform?

Ans. The convolution property for Z-transforms is very important for systems

analysis and design. In words : The transform of the convolution is the product of the transforms.

Where denotes convolution (in this case, discrete-time convolution).

Proof. This is somewhat easier (and more general) to prove for noncausal sequences.

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Q.39. State the correlation property of two sequence in z-domain. Give its ROC.

Ans. Correlation is a measure of the degree to which two signals are similar. The

correlation of two signals is divided into two.1. Cross correlation (b) Auto correlation

Q. 40. Find out the Z-transform for the following discrete time sequence.

Ans.

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Q. 41. Determine to z-transform of the following signal and sketch the pole zero

pattern:

Ans.

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Q. 42. Find z transform of

Ans. We have standard z-transform pair.

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Q. 43, What are the various methods to find out inverse z transform?

Ans. (a) Cauchy Rihemen’s theorem

(b) Long division method.

(c) Partial function.

Q. 44. Find the z-transforms of

Ans.

The X (z) is finite for all values of because

The ROC is entire z-.plane.

Q. 45. Determine the system function

Ans. Taking z-transform of both sides.

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Q. 46. Determine the pole-zero plot for the system described by difference equation

Ans. Taking z-transform of both sides.

The ROC & pole zero plot shown in Fig. below

From the following figure, we can observe the followmg

1.ROC of the system function include unit circle.

2. ROC of the system function cannot have any poles.

Q. 47. Determine the signal x(n) whose z-transform is given by

Ans. By taking the first derivative of X (z), we obtain

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Q. 48. What is the relation between z transform and Laplace transform?

Ans. The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of

Where T = 1/fs is the sampling period (in units of time e.g. seconds) and fs is the

sampling rate (in samples per second or hertz)

Let

be a sampling impulse traln (also called a Dirac comb) and

be the continuous-time representation of the sampled x(t).

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The Laplace transform of the sampled signal

This is precisely the definition of the unilateral Z-transform of the discrete function x[n]

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal:

The similarly between the z and Laplace transforms is expanded

upon in the theory of time scale calculus.

Q. 49. Find out the Z-transform for the following discrete time sequence

Ans.

UNIT II FREQUENCY TRANSFORMATIONS

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Q. 1. Define DFT.

Ans. It is a finite duration discrete frequency sequence which is obtained by sampling one period of fourier transform. Sampling is done ‘N’ equally spaced points over the period extending from . The DFT of discrete sequence x(n) is denoted by x(k) and it is given by.

where k = 0, 1, 2 N—I.

Q. 2. Define the Discrete Time Fourier Transform.

Ans. The Discrete Time Fourier Transform of a discrete line signal x(n) is expressed as

DTFT is periodic units period . So any interval of length is sufficient for the

complete specification of the spectrum. Generally, we draw the spectrum in the fundamental internal

Q. 3. What is the linearity property of DTFT?

Ans. If

According to definition of DTFT

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Comparing each summation term with definition of DTFT then we can write

Q. 4 Explain the symmetry properties of DFTs which provide basis for fast algorithms.

Ans. Most approaches for improving the efficiency of computation of DFT, exploits

the symmetry and periodicity property of i.e.

Q.5 Exploit of these two basicproperty results in computational efficient algorithms which are collectively known as FFT algorithms. Q. 6. What is zero padding in DFT?

Ans. The process of lengthening a sequence by adding zero valued samples is called

appending with zeros or zero padding. This is done to equate linear convolution units circular convolutions in case of DFT.

Q. 6. What is the importance of FF1’?

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Ans. Fast Fourier Transform (FFT) is to decompose successively the N-point DFT computation into computations of smaller size DFT’s and to take advantage of the

periodicity and symmetry properties of the complex number Such decompositions, if properly carried out, can result in a significant surving in the computational complexity given by the total number of multiplications and the total number of additions needed to compute all N DFT samples. The total no. of complex multiplications is reduced to w.r.t. DFT and the total no. of complex additions is

Q. 7. What is the quantization error in the direct computation of DFT?

Ans. The quantization errors in the direct computation of DFT is in particular the effect of round off errors due to the multiplications performed in the DFT with fixed point arithmetic, for example.

A finite duration sequence is defined as

[x(n)] is a complex valued sequence. Assume that the real and imaginary components

of x(n) and are represented by b bits. Consequently the computation of the product

x(n) W requires four real multiplications. Each real multiplication is rounded from 2b bits to b bits and hence there are four quantization errors for each complex valued multiplication.

In the direct computation of the DFT, there are N complex valued multiplications for each point in the DFT. Therefore the total number of real multiplications in the computation of a single point in the DFT is 4N. Consequently there are 4N quantization errors.

Q. 8. What is the advantage of in-place computation?

Ans. The main advantage of in-place computation is reduction in the memory size

in-place computation reduces the memory size.

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‘a’ & ‘b’ are inputs and ‘A’ and ‘B’ are outputs of butterfly. For anyone input ‘a’ and ‘b’ two memory locations are required for each. One memory location to store real part and other memory location to store imagining part. So for both inputs ‘a’ & ‘b’ = 2 + 2 = 4 memory location are required.

Thus outputs ‘A’ & ‘B’ are calculated by using the values ‘a’ & ‘b’ stored inmemory.

‘A’ & ‘B’ complex numbers, so 2 + 2 = 4 memory location are required.

Once the computation of ‘A’ & ‘B’ done then values of ‘a’ & ‘b’ are not required. Instead of storing ‘A’ & ‘B’ at other memory locations, there values are stored at the same place where ‘a’ & ‘b’ were stored. That means ‘A’ & ‘B’ are stored in the place of ‘a’ & ‘b’. This is called as in-place computation.

Q. 9. Indicate the number of stages, the number of complex multiplications at each stage, and the total number of multiplications required to compute 64-point FFT using radix-2 algorithm.

Ans.

Q. 10 Perform circular, convolution of two sequences

Ans. Circular convolution is

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Q. 11. Write application of FFT algorithm.

Ans. Linear if itering, correlation analysis and spectrum analysis are same important applications of FFT algorithm.

Q.12. Consider a complex sequence

(a) Find the Fourier Transform X(cv). (b) Find the N-point DFT

Ans. (a) Fourier transform of x (n) is given by.

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(b) N-point DFT is obtained if is replaced by

Q. 13. Compute the DFT of sequence

Ans.

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Q. 14.What is a decimation in time algorithm?DIT algorithm is used to calculate the DFT of a N point sequence. Initially the N point sequence is divided into two N/2 point sequences Xeven (n) and Xodd (n). The N/2 point DFTs of these two sequences are evaluated and combined to give the N point DFT. Similarly the N/2 point DFTs can be expressed as a combination of N/4 point DFTs. This process is continued until left with 2 point DFT. This algorithm is called decimation in time because the sequence X(n) is often splitted into smaller sequences.Q.15. Compute the DFT of x(n) =δ(n).

N-1 N-1 X(k) = ∑ x(n)WNKn = ∑ δ(n) WNKn = 1. n=0 n=0

Q.16.What is meant by radix-2 FFT?The FFT algorithm is most efficient in calculating N point DFT. If the number of point N can be expressed as a power of 2 ie N= 2M where M is an integer , then this algorithm is known as radix-2 FFT algorithm.Q.17. What is decimation in frequency algorithm?It is one of the FFT algorithms. In this the output sequence X(k) is divided into smaller subsequences, that is why the name decimation in frequency. Initially the input sequence is divided into two consisting of the first N/2 samples of X(n) and the last N/2 samples of X(n).The above procedure can now be iterated to express each N/2 point DFT as a combination of two N/4 point DFTs.This process is continued until we are left with 2 point and 1 DFT.Q.18. What are the differences and similarities between DIF and DIT algorithms?Differences:For DIT the input is bit reversed while the output is in natural order, whereas for DIF the input is in natural order while the output is bit reversed.The DIF butterfly is slightly different from the DIT butterfly, the difference being that the complex multiplication takes place after the add-subtract operation in DIF.Similarities:Both algorithms require same number of operations to compute the DFT. Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

Q.19. Explain In-place computation.To compute the elements p and q of the mth array , it is required to have elements in the p and q of the (m-1) array. If Xm(p) and Xm(q) are stored in the same register as Xm-1(p) and Xm-1(q) respectively ,it is possible to implement the above computation with only N array of complex storage registers. This kind of computation is commonly referred to as In-place computation.

Q.20.What are the applications of FFT algorithms?The applications of FFT algorithms includeLinear filtering, Correlation, Spectrum analysisQ.21.Calculate the number of multiplications needed in the calculation of DFT and FFT with 64 point sequence.Ans.Number of complex multiplications required using direct computation is

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N2 = 642 = 4096Number of complex multiplications required using FFT is(N/2) log N = ((64/2) log 64 = 192speed improvement factor (4096/192) = 21.33.Q.22. Define discrete linear convolution.The discrete convolution of the two discrete variable function x(n) and h(n) is thediscrete variable function y(n) given by the summation ∞y(n)= ∑ x(k) h(n-k) k=-∞Q.23. What are the properties of DIT FFT?1.Computation are done in place. Once a butterfly structure operation is performed on apair of complex numbers(a,b) to produce (A,B) there is no need to save the input pair(a,b). Hence we can store the results(A,B) in the same location as(a,b).2. Data x(n) after decimation is stored in reverse order.Q.24.The direct computation of DFT of a sequence X(n) requires 4N 2 real multiplicationsand N(4N-2) real additions.Q.25. The direct computation of DFT of a sequence X(n) requires N 2 complexmultiplications and N(N-1) complex additions.Q.26. What are the advantages of FFT algorithm?Fast fourier transform reduces the computation time. In DFT computation, number of multiplication is N2 and the number of addition is N(N-1). In FFT algorithm, number of multiplication is only N/2(log2N) . Hence FFT reduces the number of elements (adder, multiplier Z &delay elements). This is achieved by effectively utilizing the symmetric and periodicity properties of Fourier transform.

UNIT III IIR FILTER DESIGN

Q. 1. What is frequency wraping n Bilinear transformation?

Ans. The mapping of frequency from 1 to is approximately linear for small value of For the higher frequencies, however the relation between Q x o becomes highly non-linear. This introduces the distortion in the frequency scale of digital filter relative to analog filter. This effect is known as wraping effect.

The influence of the wraping effect on the amplitude response can be demonstrated by considering on analog filter with no. of passband centered at regular derived digital filter has some numbers of passbands but the centre frequencies and the bandwidth of higher frequencies passband in digital domain tends to be reduced.

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Q. 2. What are the conditions for distortionless transmission?

Ans. The conditions for distortionless transmission are given below.

1. Anti-aliasing filter must be used which is a low pass filter to remove high frequency noise contain in input signal. It avoids aliasing effect also.

2.Sample and hold circuit is used to keep the voltage level constant.

3.Output signal of Digital to analog converter is analog, that is a continuous signal. But it contain high frequency components. Such high frequency components are

understood. To remove these components reconstruction filter is used.

4.Amplifiers are used sometimes to bring the voltage level of input signal upto required level for distortionless transmission.

Q. 3. Show for 3rd order butterworth low pass filter the Location of its poles and zeroes in a s—plane.

Ans.

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Q.4. What are methods used to convert analog to digital filter?Approximation of derivatives, Impulse invariant method & Bilinear transformation method.Q.5. Write the pole mapping rule in Impulse invariant method?A pole located at s = sp in the s plane is transferred into a pole in the z plane located atZ = espTs

Q.6. What are the disadvantages of Impulse invariant method?Although this method is useful for implementing LPF and HPF the method is unsuccessful for implementing digital filters for which |H(jw)| does not approach zero for large value of w such as the high pass filter .Q.7. What are the advantages of Bilinear transformation method?

The Bilinear transform method provides non linear one to one mapping of the frequency points on the jw axis in the S plane to those on the unit circle in the Z plane.i.e Entire jw axis for - <w < maps uniquely on to a unit circle -/T <w/T < -/T .This procedure allows us to implement digital high pass filters from their analog counter parts.Q.8. Define the pole mapping rule in Bilinear transformation method.A pole at s = sp in the s plane is transferred into a zero at z=-1 and a pole at Z = (2+ sp

Ts)(2- sp Ts) in the z plane .Q.9. Define prewarping or prescaling.For large frequency values the non linear compression that occurs in the mapping of to w is more apparent .This compression causes the transfer function at high frequency to be highly distorted when it is translate to the w domain. This compression is being compensated by introducing a prescaling or prewarpping to frequency scale. For bilinear transform scale is converted into * scale (i.e) * =2/Ts tan (Ts/2)(prewarped frequency)

Q.10.What are the properties of FIR filter ?i. FIR filter is stable. ii. Linear phase which is accompanied by constant time or group delay.

Q.11. Define Group delay.Defined as derivative of phase with respect to frequency.

Q.12. Define phase delay.Defined as phase divided by frequency

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Q.13. What are the disadvantages of FIR filter?Long sequences of h(n)are generally required to adequately approximate sharp cut ff filters .A large amount of processing is required to realize the filter if slow convolution is used .By using FFT algorithms these filters can be designed more efficiently.

Q.14. What are the methods used to design FIR filter?1.Window Method: It involves straight forward analytical procedure however in some cases iteration is required to obtain the desired result2.Frequency Sampling: A desired frequency response is uniformly sampled and filter coefficient are then determined from these samples using the discrete fourier transform.3.Optimal or minimal design: Minimizing the maximum error between the desired and the actual frequency response by spreading the error in PB and SB.

Q.15. Why direct Fourier series method is not used in FIR filter design?The impulse response h(n) is infinite in duration. The filter is unrealizable since theimpulse response begins at- ∞ i.e no finite amount of delay can make the impulseresponse realizable. Therefore the filter which results from a Fourier seriesrepresentation of h(ejw) is an unrealizable FIR Filter.

Q.16. Comparison of analog and digital filters.Analog filter Digital filter1. In analog filter both input and outputcontinuous time signal

1. In digital filter,both the input and output are discrete time signals.

2. It can be constructed using active and passive components.

2. It can be constructed using adder, multiplier and delay units.

3. these filters operate in infinite frq. Range, theoretically but in practice it is limited by finite max. operating freq. depending upon the devices used.

3. freq. range is restricted to half the sampling range and it is also restricted by max. computational speed available for particular application.

4. It is defined by linear differential eqn.

4.It is defined by linear difference eqn

Q.17. What are the advantages of digital filter?1. Filter coefficient can be changed any time thus it implements the adaptive future.2. It does not require impedance matching between input and output.3. Multiple filtering is possible. 4. Improved accuracy, stability and dynamic range.

18. What are disadvantages of Digital Filter?The bandwidth of the filter is limited by sampling frequency.The performance of the digital filter depends on the hardware used to implement the filter.The quantization error arises due to finite word length effect in representation of signal and filter coefficient.

Q.19. What is the difference between Chebyshev Filter type I and typeII?Filter TypeI:It is all pole filter and exhibits equiripples in the pass band and monotonic characteristics in the stop band.Filter Type II:It contains both poles and zeros and exhibits a monotonic behaviour in

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the pass band and equiripple in the stop band.Q.20. What are the properties of chebyshev filter?1. For ω ≥ 1 H(jω) decreases monotonically towards zero.2. For ω ≤ 1 H(jω) it oscillates between 1 and 1\(1+^2)Q.21. Compare Butterworth filter and chebyshev filter.Butterworth filter1.The Magnitude response of Butterworth filter decreases monotonically as the frequency increases.2. The Transition width is more3.The order of butterworth filter is more, thus it requires more elements to construct and is expensive.4. The Poles of the butterworth filter lies along the circle.5. Magnitude response is flat at ω=0 thus it is known as maximally flat filter.Chebyshev Filter1.The Magnitude response of Chebyshev filter will not decrease monotonically with frequency because it exhibits ripples in pass band or stop band.2. The Transition width is very small3. For the same specifications the order of the filter is small and is less complex and inexpensive.4.The poles of chebyshev filter lies along the ellipse.5.Magnitude response produces ripples in the pass band or stop band thus it is known as equripple filter.Q.22. Compare Bilinear Transformation and Impulse Invariant

TransformationBilinear Transformation Impulse Invariant Transformation1. It is one to one mapping 1. It is many to one mappping2. The relation between analog and digital frequency is nonlinear, ie Ω=2/T tan( ω/2)

2.The relation between analog and digital frequency is linear,ie ω=ΩT or Ω=ω/T

3. Due to nonlinear relation between ω and Ω distortion occurs in frequency domain of digital filter.

3. The aliasing error occur due to sampling thus this method is suitable for design of only band limitied filters such Low pass and Band pass.

4. Due to the warping effect both amplitude and phase response of analog filter are affected but the magnitude response may be preserved by applying prewarping procedure.

4. The frequency response of analog can be preserved by selecting low sampling time or high sanpling frequency.

UNIT IV FIR FILTER DESIGN

Q.1. What is the basic difference between cascade form and direct form structures for FIR systems?

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Ans. Basis occurs in the usage of memory space in bats coses. Cascade form is basically in need of series memory. No of memory space required less in case of direct-2 form of FIR w.r.t. cascade form start use of FIR systems.

Q. 2. Which is more sensitive network to finite word length?

(a) Direct form-ll

(b) Cascade form

Justify your answer.

Ans. The direct form II realization requires only the layer of M or N storage elements. When compared to direct form I realisation the direct form II uses minimum number of storage elements and hence said to be a Canonic striictur JJrweves wJ’ the Jc is performed sequentially, the direct form II needs two adders instead of one adder required for the direct form I.

Though the direct form I and II are commonly employed, they have two drawbacks viz (i) they lock hardware flexibility and (ii) due to finite precision arithmetic, the sensitivity of the co-efficients to quantisation effects increases with the order of the filter. This sensitivity may change the co-efficient values and hence the frequency response, thereby causing the filter to become unstable. To overcome these effects, the cascade and parallel realizations can be implemented.

Q. 3. Compare different form structures of filter realization from the point of view of speed and memory requirement.

Ans. The structural representation provides the relations between some pertinent internal variable with the input and output that in turn provide the keys to implementations. There are various form of structural representations of a digital filter.

In digital implementations, the delay operation can be implemented by providing stronger register for each unit delay that is required.

In case of direct I form structure realization separate delay for both input and output signal samples. So more memory is utilized by this form. For example.

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In case of direct-Il form structure realization only one delay is required for both input and output signal samples. Therefore it is more efficient in term of memory requirements. For example:

Q. 4. What are the finite word length effects or error in filters?

Ans. The following are some of the finite word length effects in digital filters:

1. Errors due to quantization of input data by A/D converter.

2. Errors due to quantization of filter co-efficient.

3. Errors due to rounding the products in multiplication.

4. Errors due to overflow in addition.

5 Limit cycles

Q 5 What is rounding effect’

Ans. Rounding is the process of reducing size of a binary number to finite size of ‘b’ bits such that the rounded b-bit number is closest to the original unquantized number.

The rounding process consists of truncation and addition. In rounding of a number to b-bits, first the unquantized number is truncated to b-bits by retaining the most significant b-bits Then zero or one is added to the least significant bit of the truncated number depending on the bit that is next to the least significant bit that is retained.

For Example:

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0.101010 rounded to four bits is either

0.1010 or 0.1011 (Here adding one is called rounding up).

Error due to rounding: The quantization error is fixed point number due to rounding is defined as

The range of error due to rounding for all the three formats (i.e. one’s complement, two’s complement and sign-magnitude) of fixed point presentation is same.

In fixed point representation the range of error made by rounding a number to ‘b’ bits is

Relative error due to rounding of a floating point number is given by:

Q. 6. What is fixed point representation?

Ans. In fixed point representation the bits allowed for integer part and fractional part and so the position of binary point is fixed. The main drawback of this representation is that, due to the fixed integer and fraction part, too large and to small values cannot be represented. The bit to the right represent the fractional part of the number and those to the left represent the integer part.

For Example:

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The binary no. 010.11100 has the value 2.875 is decimal.

The negative numbers are represented in three different form for fixed point arithmetic

1. Sign-magnitude form.

2. One’s-complement form

3. Two’s-complement form.

1. Sign Magnitude form: In this form, the MSB is used to represent the given no. is positive or negative. Let ‘N’ be the length of binary bits, then (N-i) bit will represent magnitude and MS represent sign.

For example:

2. One’s complement form: In this form the positive number is represented as in the sign magnitude notation. But the negative number is obtained by complementing all thebits of the positive number.

For example:

3. Two’s complement form: In this form positive numbers are represented as in sign magnitude and one’s complement. The negative number is obtained by complementing all the bits of the +ve number and adding one to the least significant bit.

For example:

Q. 7. What is floating point representation?

Ans. In floating point representation, a positive number is represented as

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Where M is called mantissa and it will be in binary fraction format. The value of M will be in the range of and E is called exponent and it is either a positive or negative integer.

In this form, both mantissa and exponent uses one bit for representing sign. Usu

- ally the LSB is mantissa and exponent is used to represent the sign. A ‘I’ in the LSB represent negative sign and a ‘0’ in the LSB represent positive sign.

The floating point representation is explained by considering a five bit mantissa and three bit exponent with a total size of eight bits. In mantissa the LSB is used to represent the sign and other four bits are used to represent a binary fraction number. In exponent the LSB is used to represent the sign and the other two bits are used to represent a binary integer number.

Q 8 What is windowing?

Ans. One possible way of finding on FIR filter which approximates H(eJw) will be to

truncate the infinite fourier series at Abrupt truncation of the series would lead to oscillations m the pass-band and stop-band These oscillations may be reduced by use of less abrupt truncation of the fourier series. This can be achieved by multiplying the

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infinite impulse response with a finite weighing sequence w(n) called as window. This process is called as windowing.

Q 9 What is the importance of Windowing?

Ans 1. The infinite duration impulse response can be converted to a finite duration impulse response by trucating the infinite series at But this results in undesirable oscillations in the pass-band and step-band of the digital filter. This is due to slow convergence of the Fourier series near the point of discontinuity. These undesirable oscillations can be reduced by using a set of time limited weighing functions z e referred as windowing function.

2 The windowmg function consists of main lobe which contains most of the energy of window function and side lobes which decay rapidly

3 A major effect of windowing is that the discontinuities is are converted into transition bands between values on either side of the discontinuity

4 Window function have side lobes that decrease in energy rapidly as tends to

Q.10. Compare the performance of FIR filter and IIR filter.

Ans.

Q.11. In what cases FIR filters will be preferred over IIR filters?

Ans. Table IIR and FIR characteristics comparison

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Q. 12. What will happen if length of windows is increased in design of FIR filters?

Ans. If length of window is increased in design of FIR filter more coefficients

need to be calculated. A more memory space used for it. More lengths of window means more accuracy in transation process.

Q. 13. What is Gibb’s phenomenon?

Ans. Let us consider the example of a lowpass filter having desired frequency

response as depicted of in figure (a). This response has the cut off frequency at wc.

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Note : In figure oscillations or ringing takes place near band-edge of the filter. These oscillations or ringing is generated because of sidebes m the frequency response of the window function This oscillatory behaviour (i e ringing effect) near the band edge of the filter is known as Gibbs phenonzenon Thus, the ringing effect takes place because of sidelobes in W(w). These sidelobes are generated because of abrupt discontinuity (in case of rectangular window) of the window function. In case of rectangular window, the sidelobes are larger m size because the discontinuity is abrupt Therefore, ringing effect is maximum in rectangular window

Therefore, different window functions are developed which consists of taper and decays gradually toward zero This reduces sidelobes and hence ringing effect in H(w)

Q. 14. What are the essential features of a good window for FIR filters?

Ans. Features of a good window for F/R filters:

1. Side lobe level should be small.

2. Broaden middle section.

3. Attenuation should be more.

4. Smoother magnitude response.

5. The trade off between main lobe widths and side lobe level can be adjusted.

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6. Smoother ends.

7. If cosine term is used then side lobes are duced further.

Q. 15. Why FIR digital filters cannot have linear phase?

Ans. For FIR filter unit impulse response for symmetric system are given by:

If h (n) is symmetric then filter is symmetric. For antisymmetri’c sequence.

i.e. condition for linear phase.

For FIR filters m is finite i.e. may be odd, symmetric and antisymmetric conditions so in hR filters m is infinite.

So it does not satisfies linear phase condition of eq. (1) and (2). Due to ‘rn’ tends to ‘00’ hR filters cannot have linear phase.

Q. 16. What is FIR?

Ans. FIR is a finite impulse response. FIR system has an impulse response that is zero outside of same finite time interval. FIR system has a finite memory of length M samples.

Q. 17. Define Ripple ratio The Ripple ratio is defined as , the ratio of maximum sidelobes amplitude to the

mainlobe amplitude. i.e. %RR=(maximum side lobe amplitude/main lobe amplitude)x100Q.18.What is Gibb’s Oscillation?The truncation of Fourier series is known to introduce the unwanted ripples in the frequency response characteristics H(w) due to non uniform convergence of Fourier series at a discontinuity .These ripples or oscillatory behaviour near the band edge of the filter is known as “Gibb’s phenomenon or Gibb’s oscillation “.Q.19.What are the methods used to reduce Gibb’s phenomenon?There are two methods to reduce Gibb’s phenomenon1.The discontinuity between pass band and stop band in the frequency response is avoided by introducing the transition between the pass band and stop band.2.Another technique used for the reduction of Gibb’s phenomenon is by using window function that contains a taper which decays towards zero gradually instead abruptly.

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Q.20.What is meant by (zero input) limit cycle oscillation?Ans. For an IIR filter implemented with infinite precision arithmetic the output should approach zero in the steady state if the input is zero and it should approach a constant value if the input is a constant. However , with an implementation using a finite length register an output can occur even with zero input. The output may be a fixed value or it may oscillate between finite positive and negative values. This effect is referred to as (zero input) limit cycle oscillation.Q.21.What are the assumptions made concerning the statistical independence of various noise sources that occur in realizing the filter?Ans. Assumptions1. for any n , the error sequence e(n) is uniformly distributed over the range2. (-q/2) and (q/2). This implies that the mean value of e(n) is zero and its variance is3. The error sequence e(n) is a stationary white noise source.4. The error sequence e(n) is uncorrelated with the signal sequence x(n).Q.22. What is the difference between fixed point arithmetic and floating point arithmetic?

Fixed point arithmetic Floating point arithmetic 1. Fast operation Slow operation 2. Small dynamic range Increased dynamic range

3 Relatively economical More expensive due to costlier hardware4. Round-off errors occur Round-off errors can occur withonly in addition both multiplication and addition5. Overflow occurs in addition Overflow does not arise6. Used in small computers Used in larger general purpose

ComputersQ.23.What are the 3 quantization errors due to finite word length register in digital filters?Ans. 1. Input quantization error 2. Coefficient quantization error 3. Product quantization errorQ.24.Explain briefly the need for scaling in the digital filter implementation.Ans. To prevent overflow , the signal level at certain points in the digital filter must be scaled so that no overflow occurs in the adder.Q.25.What is limit cycles due to overflow? Or What is overflow oscillations?Ans. The addition of two fixed point arithmetic numbers cause overflow when the sum exceeds the word size available to store the sum. This overflow caused by adder make the filter output to oscillate between maximum amplitude limits. Such limit cycles have been referred to as overflow oscillations.Q.26. Define ‘dead band’ of the filter.Ans. The limit cycles occur as a result of quantization effect in multiplication. The amplitudes of the output during a limit cycle are confined to a range of values called the dead band of the filterQ.27. Express the fraction (7/8) and (-7/8) in sign magnitude, 2’s complement and

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1’s complement.Ans. fraction (7/8) = (0.111) in sign magnitude , 1’s complement and 2’s complementFraction (-7/8) = (1.111) in sign magnitude= (1.000) in 1’s complement= (1.001) in 2’s complementQ.28. The filter coefficient H = -0.673 is represented by sign magnitude fixed point arithmetic. I the word length is 6 bits , compute the quantization error due to truncation.Ans. (0.673) = (0.1010110…)(-0.673) = (1.1010110…)after truncating to 6 bits we get(1.101011) = -0.671875Quantization error = xq – x= (-0.671875)-(-0.673)= 0.001125Q.29. Give the expression for the signal to quantization noise ratio and calculate the improvement with an increase of 2 bits to the existing bit.Ans. SNR = 6b – 1.24dB,With an increase of 2 bits, increase in SNR is approximately 12dB.Q.30. Why rounding is preferred over truncation in realizing digital filters?Ans. 1. The quantization error due to rounding is independent of the type of arithmetic.2. The mean of rounding error is Zero. 3. The variance of rounding error signal is low.Q.31.What is product quantization error? Or What is round-off noise error?Ans. Product quantization error arises at the output of a multiplier. Multiplication of a ‘b’ bit data with a ‘b’ bit coefficient results in a product having 2b bits. Since a ‘b’ bit register is used , the multiplier output must be rounded or truncated to ‘b’ bits which produces an error. This error is known as product quantization error.

UNIT V APPLICATIONS

1. What is multi-rate signal processing?The process of converting a signal from a given rate to a different rate is called sampling rate conversion. Systems that employ multiple sampling rates in the processing of digital signals are called multi rate digital signal processing.

1. Define down-sampling.The process of reducing the sampling rate by an integer factor(D) is called decimation of the sampling rate. It is also called down sampling by factor(D).Decimator consists of decimation filter to band limit the signal and down sampler to decrease the sampling rate by an integer factor (D).

2. What is meant by up-sampling?Increasing sampling rate of a signal by an integer factor I is known as Interpolation or up-sampling. An increase in the sampling rate by an integer factor I may be done by interpolating (I-1) new samples between successive values of the signals.

3. If the spectrum of a sequence x(n) is X(ejw),then what is the spectrum of a signal down-sampled by a factor 2.y(n) = 1/2 [1 + ejπn/2]

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4. If the Z transform of a sequence x(n) is X(z), then what is the Z- transform of a sequence down sampled by a factor M?Y(z) = Σ v(m) [1/M Σ ej2πmk/D] z-m/D

5. What must be done to avoid aliasing?(i) Pre alias filter must be used to limit band of frequencies of the signal

to fm Hz.(ii) Sampling frequency ‘fs’ must be selected such that fs > 2 fm

6. What is the need for anti aliasing filter prior to down sampling?Anti aliasing filter is used to avoid aliasing caused by down sampling the signal x(n).

7. What is the need for anti imaging filter after up sampling a signal?Anti imaging filter removes the unwanted images that are that are yielded by up sampling.

8. Mention the areas in speech processing?Encoding, Synthesis and recognition.

9. Give some applications of speech synthesis.(i) Automatic Intercept system for telecommunications(ii) Announcement systems about time, weather warnings etc.(iii) Voice output of electronic mail, voice alarms(iv) Reading machines for dumb or blind(v) Data base enquiry services in railways, flight information, stock prices,

quotations, sports scores etc.10. Give some applications of multi rate signal processing.

1. Design of phase shifters2. Interfacing of digital systems with different sampling rates3. Implementation of narrow band LPF & implementation of Digital Filter Bank4. Sub band coding of speech signals & Quadrature mirror filter5. Trans multiplexers & Over sampling of A/D and D/A conversion

11. Mention the applications of speech coding.Digital transmission like telephony, narrow band cellular radio, military communications and secrecy missions, voice mail sent on telephone networks, voice encryption, integrated voice and data transmission over packet networks.

12. Give the main classification of speech sounds.Voiced sounds and unvoiced sounds

13. Give some methods of analysis of speech in DSP.Short time Fourier analysis, Homomorphic Filtering and Linear prediction.

14. Give the range of sampling rate for speech coding.Sampling rate : 8 KHz

15. What are the methods of speech coding techniques?Waveform coding:(i)Pulse code modulation (ii) Adaptive pulse code modulation (iii)Linear Predictive coding (iv) Frequency domain coding.

16. What is sub-band coding?The speech signal is applied to an analysis filter bank consisting of a set of Q band pass filters. This digital filtration divides the speech signal into a non overlapping frequency bands. These filter banks are contiguous in frequency. Hence, by additive

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recombination of the set of sub band signals, one can approximately generate the original speech signal.

17. What is meant by Image Enhancement?Image enhancement is to improve the appearance of images for human perception by making some features of the image like edges or contrast, more prominent relative to others. It is done for the purpose of image analysis or for display

18. What is the use of adaptive filters?Adaptive filters are capable of adjusting their co-efficient continuously during transmission of data and this is done by operating on the received signal in accordance with some algorithm. These filters are used for adaptive equalization of channel output and adaptive prediction in adaptive Differential pulse code modulation.

19. Give the advantages of digital recording.1. A high signal to noise ratio limited by A to D conversion accuracy2. Absence of wow- flutter speed variation3. Elimination of harmonic distortion at upper signal extremity4. Removal of amplitude variations caused by changes in tape magnetization5. Avoiding inter channel cross talk

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Chapter 1 : Introduction · Web viewWhat is the relationship between Z transform and the Discrete...

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