DSS - QB Solutions Part 1

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Digital Signals and Systems – QB Solution

Unit I

1. What is signal processing?

Signal processing is a method of extracting information from the signal which in turn depends on the type of signal and the nature of information it carries. Thus signal processing is concerned with representing signals in mathematical terms and extracting the information by carrying out algorithmic operations on the signal. The information contained in the signal can also be extracted either in the original domain or in the transformed domain.

Two Types of Signal Processing

- Analog Signal Processing(ASP): The processing of analog signals to extract information is called as ASP

-- Digital Signal Processing: The processing of analog signals to extract information is

called as DSP

2. What are the advantages of Digital Signal Processing (DSP) over Analog Signal Processing (ASP)?

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3. What are the disadvantages of DSP over ASP?

4. Discuss the applications of DSP?

5. Define & give the graphical representation of the following functions: i. Unit ramp


Ideal for B.Sc.IT / Com. Sci. www.bepioneer.com 99208 12021 / 2808 2086ii. Unit step

iii. Unit impulse



5. Discuss the classification of Signals.

Signals can be classified based on their nature and characteristics in the time domain. They are broadly classified as

(i) Continuous-time signals A continuous-time signal is a mathematic ally continuous function and the function is defined continuously in the time domain.

(ii) Discrete-time signals. Discrete-time signal is specified only at certain time instants. The amplitude of the discrete-time signal between two time instants is just not defined. Figure below shows typical continuous-time and discrete-time signals.

/Both continuous-time and discrete-time signals are further classified

(i) Deterministic and non-deterministic signals.(ii) Periodic and aperiodic signals.(iii) Even and odd signals.(iv)Causal and Non-causal Signal (v) Energy and power signals.

Note: Also, write down the one line definitions of the each of the above type of signals in the answer



6. 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

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

i. Shifting

ii. Folding

iii. Time scaling

Shifting In the case of discrete time signals, the independent variable is the time, n. A signal x(n) may be shifted in time, i.e. the signal can be either advanced in the time axis or delayed in the time axis. The shifted signal i represented by x(n — k), where k is an integer. If ‘k’ is positive, the signal is delayed by k units of time and if k is negative, the time shift results in an advance of signal by k units of time. In realtime signal processing applications, the operation of advancing the time base of the signal is physically unrealizable.



Folding This operation is done by replacing the independent variable n by —n. This results in folding of the signal about the origin, i.e. ii = 0. Folding is also known as the reflection of the signal about the time origin n = 0. Folding of a signal is done while convoluting the signal with another.

Time scaling This involves replacing the independent variable n by kn, where k is an integer. This process is also called as down sampling. If x(n) is the discrete-time signal obtained by sampling the analog signal, x(t), then x(n) = x(nT ), where T is the sampling period.



9. Discuss the classification of systems.

Ans:

Note: Also, write down the one line definitions of the each of the above type of system in the answer


Ideal for B.Sc.IT / Com. Sci. www.bepioneer.com 99208 12021 / 2808 208610. Draw and explain the block diagram of an analog – to – digital converter.

Ans:

Figure 1.11 shows the block diagram of an analog-to-digital converter. The sampler extracts the sample values of the input signal at the sampling instants. The output of the sampler is the discrete4ime signal with continuous amplitude. This signal is applied to a quantiser which converts this continuous amplitude into a finite number of sample values. Each sample value can be represented by a digital word of finite word length. The final stage of ana1og-todigital conversion is encoding. The encoder assigns a digital word to each quantised sample

Sampling of Continuous-time Signals

Sampling a signal is the process of acquiring its values only at discrete points in time.

Sampling is a process by which a continuous-time signal is converted into a discrete time signal. This can be accomplished by representing the continuous time signal x(t), at a discrete number of points. These discrete number of points are determined by the sampling period, T, i.e. the samples of x(t) can be obtained at discrete points t = nT, where n is an integer.

Quantisation and encoding

A discrete-time signal with continuous-valued amplitudes is called a sampled data signal, whereas a continuous time signal with discrete-valued amplitudes is referred to as a quantised boxcar signal. Quantisation is a process by which the amplitude of each sample of a signal is rounded off to the nearest permissible level. That is, quantisation is conversion of a discrete time continuous-amplitude signal into a discrete-time, discrete-valued signal. Then encoding is done by representing each of these permissible levels by a digital word of fixed wordlength.

The process of quantisation introduces an error called quantisation error and it is simply the difference between the value of the analog input and the analog equivalent of the digital representation. This error will be small if there are more permissible levels and the width of these quantization levels is very small. In the analog-to-digital conversion process, the only source of error is the quantiser. Even if there are more quantisation levels, error can occur if the sina1 is at its maximum or minimum value for significant time intervals. Figure 1.17 shows how a continuous-time signal is quantised in a quantiser that has 16 quantising levels



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

Ans:

Sampling a signal is the process of acquiring its values only at discrete points in time.

Sampling is a process by which a continuous-time signal is converted into a discrete time signal.

Sampling by Impulse Function

The sampling function g(t), discussed above, was periodic. The pulse width of the sampling function must be very small compared to the period, T. The samples in digital systems are in the form of a number, and the magnitude of these numbers represent the value of the signal x(t) at the sampling instants. In this case, the pulse width of the sampling function is infinitely small and an infinite train of impulse functions of period T can be considered for the sampling function. That is,

The sampling function as given in Eq.1.49 is shown in Fig.1.14. When this sampling function is used, the weight of the impulse carries the sample value. The sampling function g(t) is periodic and can be represented by a Fourier series as in Eq.L41, which is repeated here.



12. What is meant by quantisation and encoding?

Ans:

A discrete-time signal with continuous-valued amplitudes is called a sampled data signal, whereas a continuous time signal with discrete-valued amplitudes is referred to as a quantised boxcar signal. Quantisation is a process by which the amplitude of each sample of a signal is rounded off to the nearest permissible level. That is, quantisation is conversion of a discrete time continuous-amplitude signal into a discrete-time, discrete-valued signal. Then encoding is done by representing each of these permissible levels by a digital word of fixed wordlength.

The process of quantisation introduces an error called quantisation error and it is simply the difference between the value of the analog input and the analog equivalent of the digital representation. This error will be small if there are more permissible levels and the width of these quantization levels is very small. In the analog-to-digital conversion process, the only source of error is the quantiser. Even if there are more quantisation levels, error can occur if the sina1 is at its maximum or minimum value for significant time intervals. Figure 1.17 shows how a continuous-time signal is quantised in a quantiser that has 16 quantising levels.



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

Ans:



14. Write a note on Dirichlet’s conditions.


Ideal for B.Sc.IT / Com. Sci. www.bepioneer.com 99208 12021 / 2808 2086Ans:

\

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



16. Explain the following properties of Fourier Transform

i. Linearity

ii. Symmetry

iii. Scaling

Ans:



17. Find the Fourier transform of Unit Step Function.

Ans: Refer your notebook for the answer to this question

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



19. Obtain exponential Fourier series for the following waveform,



20. Simple Problems to be solved for finding

i. Whether given signal is periodic or not

ii. Whether given signal is energy or power signal etc


Ideal for B.Sc.IT / Com. Sci. www.bepioneer.com 99208 12021 / 2808 2086Ans: Refer your notebook for the sample problems to this question

Unit II:

1. State and explain Laplace Transform and its inverse transform.



2. What is region of convergence?



3. Find the Laplace transform of

i. Unit step function

ii. Sine function

iii. Cosine function

v. Rectangular pulse

vi. Sawtooth pulse



v. Rectangular pulse

vi. Sawtooth pulse

4. Discuss initial value and final value theorems in Laplace transform domain.



5. Find Laplace transform of the periodic rectangular wave form with period 2T



6. Find Laplace transform of the periodic sawtooth waveform with period of one cycle T



7. State any five properties of Laplace transform.

Ans: Refer your notebook for answer this question

8. Obtain Laplace transform for step and Impulse Responses of

i. Series R-L Circuit

ii. Series R-C Circuit

Ans:



9. Define the network transfer function & explain how to obtain output impulse & step response using transfer function.

Ans: Network transfer function is given by



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



Simple Problems to be solved

Ans: Refer your notebook for answer this question

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DSS - QB Solutions Part 1

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