# Digital Signal and Systems

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### Sample questions and answers of Digital Signals and Systems

Sample questions and answers of Digital Signals and Systems

Unit I:

**1. ****What is signal processing? **

Ans.

A signal as referred to in communication systems, signal processing, and is a function that conveys information about the behavior or attributes of some phenomenon”.

Signal processing is an area of that deals with operations on or analysis of signals, or measurements of time-varying or spatially varying physical quantities.

Signals of interest can include sound, electromagnetic radiation, images, and sensor data,

for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals, and many others.

**. 2. What are the advantages of digital over analog signal processing?**

** **

**Ans.** The advantages of digital over analog signal processing are given below

Accuracy: To design analog system analog components like resistors, capacitors and inductors are used. The tolerance of these components reduce accuracy of analog system. While in case of DSP, much better accuracy is obtained.

Versatility: Digital system can be reprogrammed for other applications (where programmable DSP chips are used). Moreover digital systems can be ported to different hardware

Repeatability Digital systems can be easily duplicated. These systems do not depend upon component tolerances and temperature

Simplicity: It is easy to build any digital system as compared to an analog one.

Easy up gradations: Because of use of software, digital signal processing systems can be easily upgraded compared to analog system

Compatibility: In case of digital systems, generally all applications need standard hardware. Thus operations of dsp system are mainly dependent on software. Hence universal compatibility is possible compared to analog systems

Cheaper In many applications, the digital systems are comparatively cheaper than analog systems.

Remote processing: Analog signals are difficult to store because of problems like noise and distortion while digital signal can be easily stored on storage media like magnetic tapes, disks etc Thus compared to analog signals, digital signals can be easily transposed so remote processing of digital signal can be done easily.

Implementation of algorithms: The mathematical processing algorithms can be easily implemented in case of digital signal processing but such algorithms are difficult to implement m case of analog signals

** 3. 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

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 at least 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.

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.

DSP systems are expensive as compared to analog system.

**4. What are the applications of DSP?**

** **

**Ans.** The applications of DSP are given below

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

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

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

Biomedical like scanners ECG analysis, patient monitoring

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

Military like Sonar processing, radar processing, secure communication.

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

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

5. **Define & give the graphical representation of the following functions:**

- Unit ramp
- Unit step
- Unit impulse

**Ans :**

** **

** **

** **

** **

- Unit ramp function

f (t)

t

0 t

f (t)=t, t 0;

=0, else

- Unit step function

f (t)

f (t)=1, t 0 1

=0, else

- Unit impulse function

f (t)=1, t=0 f(t)

=0, else 1

0 t

6. Discuss the classification of Signals.

Ans: Some important classifications of signals

- Analog vs. Digital signals: A signal with a magnitude that may take any real value in a specific range is called an analog signal while a signal with amplitude that takes only a finite number of values is called a digital signal.
- Continuous-time vs. discrete-time signals: continuous-time signals may be analog or digital signals such that their magnitudes are defined for all values of
*t*, while discrete-time signal are

analog or digital signals with magnitudes that are defined at specific instants of time only and are undefined for other time instants.

- Periodic vs. aperiodic signals: periodic signals are those that are constructed from a specific shape that repeats regularly after a specific amount of time
*T*_{0}, [i.e., a periodic signal*f*(*t*) with period*T*_{0 }satisfies*f*(*t*) =*f*(*t*+*nT*_{0}) for all integer values of*n*], while aperiodic signals do not repeat regularly. - Deterministic vs. probabilistic signals: deterministic signals are those that can be computed beforehand at any instant of time while a probabilistic signal is one that is random and cannot be determined beforehand.

7. Show that the product of two even signals or two odd signals is an even signal and that the product

- of an even and an odd signal is an odd signal.

Ans : Let x(t)= (t). (t)

If (t) and (t) are the two even signals, then

(t)= (-t) and (t)= (-t)

x (t)= (t) (t)= (-t). (-t)=x (t)

It is proved that, the product of two even signals is an even signal.

If (t) and (t) are the two odd signals, then

(-t)=- (t) and (-t)=- (t)

x (t)= (-t) (-t)= (t). (t)}= (t). (t)=x (t)

It is proved that, the product of two odd signals is an even signal.

If (t) is an even signal and (t) is an odd signal, then

(-t)= (t) and (-t)=- (t)

x (t)= (-t) (-t)= (t). (t)}= (t). (t)=-x (t)

It is proved that, the product of an odd signal and an even signal

Is an odd signal

8. With illustrations, explain the following for discrete – time signals.

i. Shifting

ii. Folding

iii. Time scaling

9. Discuss the classification of systems.

Ans: The systems are classified into following types

- Static and dynamic systems

Static system is a system that the output of the system depends on the input at the particular time. It does not depend on past or future values of the input.

It doesn’t contain any storage elements.

The input/output relation of such systems doesn’t involve integrals or derivatives.

Dynamic system is a system that the output of the system depends on the input at the particular time and also other times. It depends on past or future values of the input.

- Linear and Non-linear systems

A system is said to be linear system, if it follows the following equation.

H [ax (t) +by (t)] = a H[x (t)] +b H[y (t)]

Where, x (t) and y (t) are the two inputs & ‘a’ and ‘b’ are the two constants.

- Time-variant and time-invariant systems

A system is said to be time-invariant if, the input-output relationship doesn’t with time. It is also

called fixed system. The condition for a system to be fixed is

H[x (t-T)] =y (t-T)

If a system doesn’t follow the above equation, the system is called time-variant system.

- Causal and non-causal system

A system is said to be causal if, the output depends on the past and present inputs but not on the future inputs.

A system is said to be Non-causal if, the output depends on the past and present inputs and on the future inputs also.

- Stable and unstable systems

A system is said to be stable if it follows the following conditions

i). If the system transfer function is rational function, the degree of the numerator must be no larger than the degree of denominator.

ii). the poles of the system must lie on the left half of the s-plane or within the unit circle.

iii). If a pole lies on the imaginary axis, it must be a single-order one i.e. no repeated poles must lie on the imaginary axis.

A system is said to be unstable, if it doesn’t follow the above conditions.

10. Draw and explain the block diagram of an analog – to – digital converter.

11. What is meant by sampling? State sampling theorem?

Ans : It is defined as the sampling frequency (f_{s}) should be greater than or equal

to twice the maximum frequency(fm) then we can generate the original signal if the condition does not satisfy we get the signal in the distorted manner.

Sampling theorem is given as, fs 2.fm

12. What is meant by quantization and encoding?

Quantization, involved in image processing, is a lossy compression technique achieved by compressing a range of values to a single quantum value.

When the number of discrete symbols in a given stream is reduced, the stream becomes more compressible.

For example, reducing the number of colors required to represent a digital image makes it possible to reduce its file size.

13. Write down the trigonometric form of the Fourier series representation of a periodic signal.

14. Write a note on Dirichlet’s conditions.

Fourier series exists only when the function f(t) satisfies following three conditions, i.e. called **Dirichlet’s conditions**

f (t) has a finite average value over the period T i.e. the system should not be open.

f (t) must possess only a finite number of discontinuities in the period T.

15. State and prove Parseval’s theorem for Fourier transform.

The Fourier Transform energy-conserving relation and the energy may be found from f(t) or its spectrum |F(j )| as

E= =

It is also called Rayleigh’s energy theorem.

16. Explain the following properties of Fourier Transform

i. Linearity

A system is said to be linear system, if it follows the following equation.

F [ax (t) +by (t)] = a F[x (t)] +b F[y (t)]

Where, x (t) and y (t) are the two inputs & ‘a’ and ‘b’ are the two constants

ii. Symmetry

A system is said to be symmetry, if it satisfies the following condition

F(jt)=2 .f(- )

- Scaling

If F f (t) =F (j )

Then, Ff (at) = F ( )

- Find the Fourier transform of Unit Step Function.

See the notes

- Determine the Fourier transform of Signum function & also plot the amplitude and phase spectra.

See the notes

Unit II:

- State and explain Laplace Transform and its inverse transform.

See the notes

- What is region of convergence?
- Find the Laplace transform of
- Unit step function
- Sine function
- Cosine function
- f(t) = 3+ 3t2– 6t + 4
- Rectangular pulse
- Sawtooth pulse
- Discuss initial value and final value theorems in Laplace transform domain.
- Find Laplace transform of the periodic rectangular wave form with period 2T
- Find Laplace transform of the periodic sawtooth waveform with period of one cycle T
- State any five properties of Laplace transform.
- Obtain Laplace transform for step and Impulse Responses of

i. Series R-L Circuit

ii. Series R-C Circuit

- Define the network transfer function & explain how to obtain output impulse & step response

Using transfer function.

10.

Determine poles, zeroes of F(s) & plot them graphically. Also obtain f(t) if F(s)=

Simple Problems to be solved

Unit III:

- Define z-Transform.
- Explain the use of z-Transform.
- How is z-Transform obtained from Laplace transform?
- State and explain the properties of z-Transform.
- Compare the properties of tw-sided z-transform with those of one-sided z-Transform
- What is the condition for z-Transform to exist?
- With reference to z-Transform, state and the initial and final value theorem.
- Obtain the Z-Transformation of x(n)=2nu(n-2).
- Determine the Z-Transform and the region of convergence of

x(n) =

10 Obtain the Z-Transform of x(n)=n

2

u(n).

- State the Contour-Integration Residue method to calculate Inverse Z-Transformation. Hence

- obtain Inverse Z-Transform of X(z) = .

- Obtain the Z-Transform of the sequence x(n)= {1,2,5,4,6}

Simple Problems to be solved

Unit IV:

- When a system is said to be linear?
- Define the terms
- Linearity
- Time invariance
- Causality

With reference to discrete time system

- Simple problems to check the Linearity and Causality of the signals.
- Explain briefly the Paley-Wiener criterion
- Explain stability in Linear Time Invariant system. What is the condiction for a system to be BIBO

stable?

- Check whether the following digital systems are BIBO stable
- y(n) = ax

2

(n)

- y(n) = ax(n) + b
- What is convolution? What are the properties of convolution?
- What is frequency response? What are the properties of frequency response?
- Simple examples for frequency response

For example

The output y(n) for an Linear Time Invariant system to the input x(n) is

y(n) = x(n) – 2x(n-1) + x(n-2)

Compute the magnitude and phase of the frequency response of the system for

10 Check whether the system F[x(n)]= n[x(n)]

2

is Linear and Time-Variant.

- Obtain Frequency Response for y(n) =x(n)+10y(n-1) with initial condition y(-1)=0.

Unit V:

- State and explain the properties of Discrete Fourier Series.
- Define Discrete Fourier Transform (DFT) for a sequence x(n)
- Explain any 5 properties of DFT
- State the relationship between DFT and z-Transform

5.

Determine DFT of the sequence x(n) =

- Define Discrete Time Fourier Transform (DTFT) and Inverse Discrete Time Fourier Transform

(IDTFT).Explain the difference between Discrete Fourier Transform (DFT) and Discrete Time

Fourier Transform (DTFT).

- Determine the Circular Correlation values of the two sequences x(n)={1,0,0,1} and

h(n)={4,3,2,1}.

- What are the methods used to perform Fast Convolution. Explain any one method giving all the

steps involved to perform Fast Convolution.

- Compute Linear and Circular Periodic Convolutions of the sequence x

1

(n)= {1,1,2,2} and x

2

(n)=

{1,2,3,4} using DFT.

10 Obtain X(k) for the sequence x(n)= {1,2,3,4,4,3,2,1} using Decimation-in-Time(DIT) , Fast

Fourier Transform(FFT) Algorithm.

11 Find the exponential form of the Discrete Fourier Series representation of x(n) if

12 Define Discrete Fourier Transform(DFT) and Inverse Discrete Fourier Transform(IDFT). Also

state the Complex Conjugate property and Circular Convolution property of Discrete Fourier

Transform(DFT).

13 Consider two periodic sequences x(n) and y(n) with period M and N respectively. The sequence

w(n) is defined as w(n)= x(n)+y(n). Show that w(n) is periodic with period MN. (Hint : Using

DFT).

14 Obtain X(k) for the sequence x(n)= 2

n

using Decimation-in-Frequency(DIF) , Fast Fourier

Transform(FFT) Algorithm.

Unit VI:

1 State the advantages of Digital filters.

- Explain the effects of windowing. Define Rectangular and Hamming window functions.
- Explain the procedure for designing an FIR filter using Kaiser window.

4.

What is bilinear transformation? Apply bilinear transformation to with T=0.1 s.

- Describe the Inverse Chebyshev filters.
- Obtain the system functions of normalized Butterworth filters for order N = 1 & 2.
- What are the advantages of FIR filter over IIR filters?
- What is an IIR filter? Compare its characteristics with an FIR filter.
- Write note on Butterworth filters.
- Write note on Chebyshev filters.
- Describe elliptical filters in detail.