syllabus

3.1

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERINGK.S.R. COLLEGE OF ENGINEERING: TIRUCHENGODE – 637 215.

COURSE / LESSON PLAN SCHEDULENAME : S.DEIVANAYAGI CLASS : B.E/ II-ECE-A & BSUBJECT : 07 EC403 - SIGNALS AND SYSTEMS

A). TEXT BOOKS:1. S.Poorana Chandra & B.Sasikala. “Signals and Systems”, Vijay Nicole Imprints Pvt. Ltd., Chennai

B). REFERENCES :1. AlanV.Oppenheim, Alan S.Willsky with S.Hamid Nawab, Signals & Systems, 2nd edn., Pearson Education, 1997.

C). EXTRA BOOKS :1. P.Ramesh Babu, “Digital Signal Processing”, Scitech Publications, 3rd edition.

D. LEGEND : L 1 - Lecture 1 BB - Block BoardA1 - Assignment 1 OHP - Over Head Projector T 1 - Tutorial 1 LCD - Liquid CrystalDisplay Tx 1 - Text 1 MD - Model Demo Rx 1 - Reference 1 pp - Pages

Sl.No. Lecture Hour Topics to be covered

Teaching Aid

RequiredBook No. / Page No.

UNIT – I CLASSIFICATION OF SIGNALS AND SYSTEMS

1 L 1Continuous time signals (CT signals), discrete time signals (DT signals)

BBTx 1 / pp SGL 1 to SGL 7Rx1 / pp 01 to 07Ex1 / pp

2 L 2,L 3 Step, Ramp, Pulse, Impulse,Exponential and sinusoidal BB Tx 1 / pp SGL 44 to SGL 51

Rx1 / pp 30 to 38

3 L 4, L 5,L6

Classification of CT and DT signals - periodic and aperiodic, Energy and power,even and odd

BBTx 1 / pp SGL 4 to SGL 28Rx1 / pp 11 to 14

4 L 7 Deterministic and Random signals BB Tx 1 / pp SGL 28 to SGL 29

5 L 8, L 9 Transformation on Independent variables BB Tx 1 / pp SGL 29 to SGL 44

Rx1 / pp 7 to 11

6 L 10 CT systems and DTsystems & Properties of Systems BB Tx 1 / pp SYS 1 to SYS 26

Rx1 / pp 38 to 56

7 L 11, L12, L13

Linearity,Causality,Time Invariance,stability,Invertibility and LTI Systems – problems, Signal Processing

BB Tx 1 / pp SYS 1 to SYS 26Rx1 / pp 38 to 56

UNIT- II ANALYSIS OF CT SIGNALS

8 L 14, L 15 Fourier series analysis BB Tx 1 / pp CFS 1 to CFS 27Rx1 / pp 177 to 201

9 L 18 Properties of CTFS and its problems BB Tx 1 / pp CFS 27 to CFS 33

Rx1 / pp 202 to 206

KSRCE/ECE SIGNALS AND SYSTEMS EC403

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10 L 19 Spectrum of CT signals BB Tx 1 / pp CFT 5 to CFT 15Ex1 / pp 5.27 to 5.29

11 L 20 Fourier Transform BB Tx 1 / pp CFT 1 to CFT 5Rx1 / pp 284 to 300

12 L 21 Properties of CTFT and its problems BB Tx 1 / pp CFT 15 to CFT 29

Rx1 / pp 300 to 330

13 L 22, L 23, L 24

Laplace Transform in SignalAnalysis-Properties & problens- Parseval’s Theorem

BBTx 1 / pp LT 1 to LT 37Rx1 / pp 654 to 670

14 L25 Sampling Theorem and aliasing & Signal reconstruction BB

Tx 1 / pp ST 1 to ST 6Rx1 / pp 514 to 520 & 527 to 534

UNIT- III LTI-CT SYSTEMS

15 L 27, L 28 Differential equation & Block diagram representation BB Tx 1 / pp CLS 29 to CLS 39

Rx1 / pp 117 to 124

16 L 29, L 30 Impulse response- Convolution Integral & Frequency response BB Tx 1 / pp CLS 1 to CLS 23

Rx1 / pp 90 to 102

17 L 31, L 32,

Differential equation using Fourier Methods BB Tx 1 / pp CFT 29 to CFT 33

Rx1 / pp 330 to 333

18 L 33, L 34 Differential equation using Laplace transforms BB Tx 1 / pp LT 37 to 43

Rx1 / pp 693 to 70019 L 35 State equations and Matrix BB Tx 1 / pp SV 14 to SV 15

UNIT IV ANALYSIS OF DT SIGNALS

20 L36 Spectrum of DT Signals BB Tx 1 / pp DFT 1 to DFT 13Rx1 / pp 211 to 221

21 L 37, L38 Discrete Time Fourier Transform (DTFT) and problems BB Tx 1 / pp DFT 13 to DFT 20

Rx1 / pp 373 to 384

22 L39, L40 Discrete Fourier Transform (DFT) and problems & FFT BB Tx 1 / pp FFT 1 to FFT 21

23 L41, L42Ztransform, ROC, Properties of ROC & problems – Poles and Zeros & problems

BBTx 1 / pp ZT 1 to ZT 11Rx1 / pp 741 to 748Ex1 / pp 10.29 to 10.50

24 L43, L44 Properties of Z-transform in signal analysis BB Tx 1 / pp ZT 16 to ZT 28

Rx1 / pp 763 to 774

25 L45, L46, L47 Inverse Z Transform BB Tx 1 / pp ZT 28 to ZT 38

Rx1 / pp 757 to 762UNIT V LTI-DT SYSTEMS

26 L48, L49 Difference equations & Block diagram representation BB

Tx 1 / pp DL 1 to DL 3 & DLS 41 TO DLS 54Rx1 / pp 117 to 127

27 L50, L51 Convolution sum & properties of DT-LTI system BB Tx 1 / pp DLS 3 to DLS 7

Rx1 / pp 75 to 8928 L52, L53 Z-transform analysis BB Rx1 / pp 774 to 783

29 L54, L55, L56

State variable equation and Matrix & Structure realization BB Tx 1 / pp SV 1 to SV13

Ex1 / pp 10.76 to 10.55


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UNIT I - CLASSIFICATION OF SIGNALS AND SYSTEMS

1. Define unit step,ramp and delta functions for CT.(OCT / NOV 2002)Unit step function is defined as

U(t)= 1 for t >= 00 otherwise.

Unit ramp function is defined asr(t) = t for t>=0

0 for t<0.Unit delta function is defined as

d(t) = 1 for t=0 0 otherwise.

2. What is period of the signal x(n) = 2 cos(n/4)?(APR / MAY 2003)x(n) = 2 cos(n/4). Compare x(n) with Acos(2πfn). This gives 2πfn = n/4 = > F = 1/(8π) which is not rational. Hence this is not periodic signal.

3. What is the total energy of the discrete time signal x(n) which takes the value of unity at n = -1, 0 & 1?(NOV / DEC 2003)Energy of the signal is given as E = ∑ |x(n)|2 = ∑ |x(n)|2

= | x(-1)|2 + |x(0)|2 + |x(1)|2 = 1+1+1 = 3.4. Draw the sinc function.( NOV / DEC 2003)

Sinc(x) = sinπx / πx. Sinc(0) = 1 by L-hospital’s rule. Other values can be calculated directly.

5. What is an energy signal? Check whether or not the unit step signal is an energy signal.(APR / MAY 2004).A signal is said to be energy signal if its total energy is finite and non-zero. That is 0< E<∞. E = ∫|x(t)|2 dt = ∫1. dt = ∞. Since x(t) = 1 for t ≥ 0

0 for t < 0. Since energy is not finite, unit step signal is not energy signal.

6. Clasify the following signals as, a) Periodic or non-periodic and b) energy or power signal. 1) eαn, α > 1, 2) e-j2πft. (NOV /DEC 2004)

a) Periodicity 1) eαn is non-periodic since it is exponential signal.2) e-j2πft is periodic signal since it is phasor of frequency f.

b) Energy or Power.1) Since eαn is non-periodic, let us calculate its energy,

E = ∑ | eαn |2 = ∞.This signal is neither energy nor power signal.


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2) Since e-j2πft is periodic signal, hence let us calculate its power,P =

Since power is finite, this is power signal.7. Draw the waveform x(-t) and x(2-t) of the signal x(t) = t for 0≤ t ≤ 3.

0 for t>3. (NOV /DEC 2004)

8. Define power signal (APR /MAY 2005).The signal x(t) is said to be power signal, if and only if the normalized average power p is finite and non-zero. 0<p<∞.

9. Is the signal x(t) = 2 cos(3πt)+7 Cos(9t) periodic? (APR /MAY 2005).Compare the given signal with, x(t) = A cos(2πf1t)+B Cos(2 πf2t)2πf1t = 3πt => f1 = 3/2 => T1 = 2/3.2πf2t = 9t => f2 =9/2π => T2 = 2π /9.T1 / T2 = (2/3) / (2π /9) = 3/ π which is not ratio of integers. Hence given signal is not periodic.

10. Is the system y(t) = y(t-1)+2t y(t-2) time invariant?(OCT / NOV 2002).Here, y(t-t1) = y(t-1-t1)+2t y(t-2-t1) and y(t, t1) = y(t-t1 - 1)+2t y(t-t1 - 2).Here y(t-t1) ≠ y(t,t1). This is time variant system.

11. Is the following system invertible? Given y(t) = x2(t). (NOV / DEC 2003).The system squares input. The invertible system has to take squareroot. But [-x(t)]2 = x2(t). This means output of is obtained for two inputs –x(t) as well as x(t). Hence this system is not invertible.

12. Check whether the system having input output relation y(t) = ∫x(τ)d τ is linear time invariant or not?(APR /MAY 2004)This is an integration of input. An integartion is always independent of time shift. Hence this is time invariant system.

13. Is diode a linear device? Give your reason.(NOV / DEC 2004)Diode is non-linear device since it opeartes only when forward biased. For neagtaive biased, diode does not conduct.

14. What is the periodicity of the signal x(t) = sin(100πt) + cos(150πt)? (NOV / DEC 2004)Compare the given signal with, x(t) = sin(2πf1t)+ Cos(2 πf2t)2πf1t = 100πt => f1 = 50 => T1 = 1/50.2πf2t = 150πt => f2 =75 => T2 = 1/75.T1 / T2 = (1/50) / (1/75) = 3/2. That is rational, the signal is periodic. The fundamental period will be, T = 2 T1 = 3 T2. i.e, least common multiple of T1 and T2. Here T = 1/25.


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PART-B

1. Find the fundamental period T of the following signal x(n) = cos(nπ/2) – sin(nπ/8) + 3 cos[(nπ/4) +(π/3)] (OCT / NOV 2002).

2. Find whether the following signal is periodic or not? X(n) = 5cos(6πn). (NOV / DEC 2003).

3. Define and plot the following signals 1) Ramp 2) Step 3) Pulse 4) Impulse and Exponential.(APR / MAY 2004).

4. What are the basic continuous time signals? Draw any 4 waveforms and write their equations.( NOV / DEC 2004).

5. Determine energy of the discrete time signal x(n) = (1/2)n, n≥0 & 3n, n<0.( MAY 2005)6. Verify whether the following system is linear.y(n) = x(n)+n(x(n+1)). (NOV 2002).7. Test wheter the system described by the equation y(n) = n x(n) is (i) linear (ii) shift

invariant.(NOV/DEC 2003).8. Verify the linearity , casuality and time invariance of the system, y(n+2) = a x(n +1) +

bx(n+3).(NOV/DEC 2004).

UNIT II - ANALYSIS OF CONTINUOUS TIME SIGNALS

PART A

1.What do the Fourier series coefficients represent?(OCT / NOV 2002). Fourier series coefficients represent various frequencies present in the signal. It is nothing but spectrum of the signal.

2.Define Fourier series.( OCT/NOV 2002) CT Fourier series is defined as, X(t) = ∑ X(k) ejkω

0t Where, X(k) = 1/T ∫ x(t) ejkω

0t dt

DT Fourier series is defined as, X(n) = ∑ X(k) ejkΩ

0n & X(k) = 1/N ∑ x(n) ejkΩ

0n

3.State Dirichlet conditions for Fourier series.(APR / MAY 2004) i) The function x(t) should be within the interval T0. ii) The function x(t) should have finite number of maxima and minima in interval T0. iii) The function x(t) should have at the most finite number of discontinuites in the interval T0. iv) The function x(t) should be absolutely integrable.i.e ., ∫|x(t)| dt < ∞.

4. What are the differences between Fourier series and Fourier transform?(OCT / NOV 2002,NOV/DEC 2004)

S.NO Fourier Series Fourier Transform01. Fourier series is calculated for

periodic signals.Fourier Transform is calculated for non-periodic as well as periodic signals

02 Expands the signal in time domain. Represents the signal in Frequency domain.

03. Three types of fourier series such as trigonometric, polar and complex exponential.

Fourier transform has no such types.


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5. What is the relationship between fourier transform and laplace transform?(APR / MAY 2003).X(s) = X(jω), when s = jω. This means fourier transform is same as laplace transform when s = jω.

6. State the modulation property and convolution (time) property of fourier transform.(APR / MAY 2003). Modulation property: x(t) cos(2πfct + Φ) ↔ ejΦ / 2 X(f - fc ) + e-jΦ / 2 X(f +fc ) Convolution Property: x1(t) * x2(t) ↔ X1(f) . X2(f).

7. Write the fourier transform pair for x(t). (NOV/DEC 2003) X(t) ↔ x(f) or x(t) ↔ X(ω).

8. What is the laplace transform of e-at sin (ωt) u(t).(NOV/DEC 2004) e-at sin (ωt) u(t) ↔ ω / [(s+a)2 + ω2].9. Determine the laplace transform of x(t) = e-at sin (ωt) u(t).(APR / MAY 2004). £ e-at sin (ωt) = £ { e-at {( ejωt - e-jωt) / 2j}} = (1 / 2j) £ { e-(a-jω)t - e-(a+jω)t }

= ω / [(s+a)2 + ω2] , ROC : Re (s) > -a.10. A signal x(t) = cos(2πft) is passed through a device whose input- output is related

by y(t) = x2(t). What are the frequency components in the output? (NOV/DEC 2004).

Since an input is square, y(t) = (cos 2πft)2 = [1+ cos 4πft] / 2 = 0.5 + 0.5 cos [2π(2f)t]. Thus the frequency present in the output is ‘2f’.

11. Define th fourier transform pair for continuous time signal.(APR / MAY 2005). Fourier transform : X(ω) = ∫x(t) e-jωt dt. Inverse Fourier transform : X(f) = 1/2π∫X(ω) ejωt d ω.

12. Find the laplace transform of x(t) = t e-at u(t), where a>0. (APR / MAY 2005). e-at u(t) ↔ 1 / (s+a’) ROC : Re(s) > -a. differentiation in s-domain property gives,

-t x(t) ↔ d / ds X(s)- t e-at u(t) ↔ d / ds [1 / (s+a)].t e-at u(t) ↔ 1/[s+a]2, ROC : Re(s) >-a.

13. What is meant by sampling. A sampling is a process by which a CT signal is converted into a sequence of discrete samples with each sample representing the amplitude of the signal at the particular instant of time.

14. State Sampling theorem. A band limited signal of finite energy, which has no frequency components

higher than the W hertz, is completely described by specifying the values of the signal at the instant of time separated by 1/2W seconds.

A band limited signal of finite energy, which has no frequency components higher than the W hertz, is completely recovered from the knowledge of its samples taken at the rate of 2W samples per second.

15. What is meant by aliasing. When the high frequency interferes with low frequency and appears as low then the phenomenon is called aliasing.

16. What are the effects aliasing. Since the high frequency interferes with low frequency then the distortion is generated.The data is lost and it can not be recovered.


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17. How the aliasing process is eliminated. i). Sampling rate fs _ 2W. ii). Strictly band limit the signal to ‘W’. This can be obtained by using the Low pass filer before the sampling process.It is also called as anti-aliasing filter.

18. Define Nyquist rate.and Nyquist interval. When the sampling rate becomes exactly equal to ‘2W’ samples/sec, for a given bandwidth of W hertz, then it is called Nyquist rate. Nyquist interval is the time interval between any two adjacent samples. Nyquist rate = 2W Hz & Nyquist interval = 1/2W seconds.

PART-B

1. Find the FS for the periodic signal x(t) = t, 0 ≤ t ≤1 and repeats every 1 sec.(MAY 2003).2. Determine the fourier series representation for x(t) = 2sin (2πt - 3) + sin (6πt).(DEC ‘03).3. Determine the trigonometric FS representation of the half wave rectifier O/P.( MAY ‘04).4. Find the fourier series of the signal x(t) = ∫ sin(2πf0mt). cos(2πf0nt) dt, where fo is the

fundamental frequency and m & n are any positive integer. (NOV/DEC 2004).5. Determine the trigonometric FS representation of the full wave rectifier O/P.(MAY ‘05).6. State and prove parseval’s theorem for complex exponential fourier series.(MAY 2005).7. Find the fourier transform of the signal x(t) and plot the amplitude spectrum.

x(t) = 1, -τ/2 ≤ t ≤ τ/2. & 0, otherwise. (OCT / NOV 2002).8. If x(t) ↔ X(s), derive the FT of x(t - τ) and e-at x(t) in terms of X(s). (NOV 2002).9. Prove that the spectrum of the product of two signals corresponds to the convolution of

their respective spectrums. (APR / MAY 2003).10. Find the inverse laplace transform of [2s2 + 9s – 47] / [(s+1) (s2 + 6s +25)]. (MAY 2003).11. State and prove parseval’s theorem for fourier transform.( MAY 2004 & DEC 2003).12. State and prove convolution property for fourier transform.( NOV/DEC 2003).13. Find the laplace transform of x(t) = e-b|t| for b<0 and b>0.( NOV/DEC 2003).14. Determine the fourier transform of x(t) = 1 for -1 ≤ t ≤ 1 & 0 otherwise. (MAY 2004).15. State and prove convolution theorem for laplace transform. (APR / MAY 2004).16. Prove that the convoution of two signals is equivalent to multiplication of their respective

spectrum in frequency domain. (NOV/DEC 2004).17. Find the LT of t x(t) and x(t-t0) where t0 is a constant term and x(t) ↔ X(s). (DEC 2004).18. Find the laplace transform of x(t)=δ(t)-4/3 e-t u(t)+1/3e2tu(t).19. Find the invers laplace transform of X(s)=1/((s+1)(s+2)).20. Determine the laplace transform of x(t) = 2t , 0 ≤ t ≤ 1 & 0, otherwise. (MAY 2005).

UNIT III - LTI- CT SYSTEMS

PART A

1. Give 4 steps to compute convolution integral. (OCT / NOV 2002).1) Folding 2) Shifting 3) Multiplication and 4) Integration.

2. What is the overall impulse response h(t) when two systems with impulse response h1(t) and h2(t) are in parallel and in series? (OCT / NOV 2002).

For parallel connection, h(t) = h1(t) + h2(t). For series connection h(t) = h1(t) *h2(t).

3. Define linear time invariant system. (APR / MAY 2003).The output response of linear time invariant system (LTI) is linear and time invariant.


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4. Define impulse response of a linear time invariant system. (APR / MAY 2003).Impulse response of LTI system is denoted by h(t). It is the response of the system to unit impulse input.

5. Write down the input-output relation of LTI system in time and frequency domain. y(t) = h(t) * x(t). Time domainY(f) = H(f) . X(f) Frequency Domain (or) Y(s) = H(s) . X(s) Frequency Domain. Here y(t) si the output, h(t) is impulse response and x(t) is the input.

6. Define transfer function in CT systems.(APR / MAY 2005 & NOV/DEC 2003).Transfer function relates the transforms of input and output,

H(f) = Y(f) / X(f), using fourier transform (or)H(s) = Y(s) / X(s), using laplace transform.

7. State the properties of convolution.( NOV/DEC 2003).1) Commutative property 2) Associative property 3) Distributive property.

8. What is the relationship between input and output of an LTI system? (MAY 2004).Input and output of an LTI system are related by, y(t) = ∫x(τ) h(t - τ) d τ.

9. What is the transfer function of a sysytem whose poles are at -0.3 ± j 0.4 and a 0 at -0.2?( NOV/DEC 2004).p1 = - 0.3 + j0.4, p2 = - 0.3 - j0.4.Z = -0.2

H(s) = [s-z] / [(s – p1)( s – p2)]. = (s+0.2) / [(s+0.3 - j0.4)(s+0.3+ j0.4)]. = (s+0.2) / [(s+0.3)2 +(0.4)2]. = (s+0.2) / [s2 + 0.6s + 0.25]

10. What is natural response?This is output produced by the system only due to initial conditions .Input is zero for natural response. Hence it is also called zero input Response.

11. What is zero input Response?This is output produced by the system only due to initial conditions .Input is zero for zero input response.

12. What is forced response.This is the output produced by the system only due to input .Initial conditions are considered zero for forced response.It is denoted by y(f )(t).

13. What is complete response?The complete response of the system is equal to the sum of naturalresponse and forced response .Thus initial conditions as well as input both areconsidered for complete response.

PART-B1. Find the convolution of x(t) and h(t). x(t) = 1, 0 ≤ t ≤ 2 & 0, otherwise.

h(t) = 1, 0 ≤ t ≤ 3 & 0, otherwise. (OCT / NOV 2002).2. Find the impulse response of the system. (OCT / NOV 2002).


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3. How do you represent any orbitrary signal in terms of delta function and its delayed function. (APR / MAY 2003).

4. Find the output of the system shown in the figure for the input e-2t u(t) using laplace transform. (APR / MAY 2004).

5. Determine the System response of the given differential equation y’’(t)+3y’(t)=x(t).,Where x(t)= e-2t u(t).

6. Determine the response of the system with impulse response h(t) = u(t) for the input x(t) = e-2t u(t). (APR / MAY 2004 & APR / MAY 2005).

7. Find the output of an LTI system with impulse response h(t) = δ(t - 3) for the input x(t) = cos 4t + cos 7t. (APR / MAY 2004).

8. Using laplace transform find the impulse response of an LTI system described by the differential equation [d2y(t) / dt2] – [dy(t) / dt] – [2y(t)] = x(t). (APR / MAY 2004).

9. Find and plot the magnitude spectrum of the transfer function. H(jω) = [ejω + α] / [ejω + 1 / α].( NOV/DEC 2004).

10. A) Define linear time invariant system. (NOV/DEC 2004).B) Define convolution intgral of a system. (NOV/DEC 2004).

11. What is meant by causality and stability? Derive the conditions for causality and stability.(APR / MAY 2005). 12. Determine the impulse response of the CT system described by the differential equation[d2y(t) / dt2] + 4[dy(t) / dt] +3[y(t)] = [dx(t) / dt] + 2x(t). (APR / MAY 2005). 13. Find the response of the system shown in figure for the input x(t) = δ(t) - δ(t – 1.5). Here h(t) is impulse response of the system. (NOV/DEC 2004).

UNIT-IV - ANALYSIS OF DISCRETE TIME SIGNALS

PART-A

1. What is the relation between Z transform and fourier transform of discrete time signal. (APR / MAY 2003).

X(ω) = X(z)|z=ejω. This means Z transform is same as fourier transform when

evaluated on unit circle.2. What is the Z transform of an u(n). (OCT / NOV 2002).

Z{an u(n)} = 1 / [1-a Z-1], ROC : |Z| > |a|.3. Define region of convergence with respect to Z transform. ( NOV/DEC 2003).

Region of convergence (ROC) is the area in Z plane where Z transform converges. In other words, it is possible to calculate the X(z) in ROC.

4. State initial value theorem of Z transform. (APR / MAY 2004).The initial value of the sequence is given as, x(0) = Lt X(z).

z→∞

5. What is the difference between the spectrum of the CT signal and the spectrum of the corresponding sampled signal. ( NOV/DEC 2004).

The spectrums of CT signal and sampled signal are related as, Xδ(f) = fs ∑ X(f - nfs).This means spectrum of sampled signal is periodic repeatation of spectrum of CT signal.It repeats at sampling frequency fs and amplitude is also multiplied by fs.

6. State final value theorem for Z transform. (APR / MAY 2005).The final value of the sequence is given as, x(∞) = Lt (1 – z-1)X(z).

z→1


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7. Define DTFT pair. (APR / MAY 2004).DTFT, X(ω) = ∑ x(n)e-jωn & IDFT, x(n) = 1 / 2π ∫ X(ω) ejωn dω.

8. State and prove time eshifting property of z transform. ( NOV/DEC 2003). Time shifting property x(n - k) ↔ z-k x(z).

9. Define one sided and two sided z transform. ( NOV/DEC 2004). ∞

One sided z transform is given as X(z)=∑ x[n] z-n. n=0

∞ Two sided z transform is given as X(z)=∑ x[n] z-n.

n= -∞10. Define Z transform.

The Z transform of a discrete time signal x[n] is denoted by X(z) and it isgiven as X(z)=_x[n] z-n.and the value n range from -_ to +_. Here ‘z’ is thecomplex variable.This Z transform is also called as bilateral or two sided Ztransform.

11. What is the differentiation property in Z domain.x[n] ↔ X(Z) thennx[n] ↔ -z d/dz{X[Z].}.

12. State the methods to find inverse Z transform.a. Partial fraction expansion b. Contour integration c. Power series expansiond. Convolution method.

13. State parseval’s relation for Z transform.If x1[n] and x2[n] are complex valued sequences, then the parseval’s relationstates that ∑x1[n] x2*[n]= 1/2j ∫X1(v). X2*(1/v*)v-1dv.

14. Define Twiddle factor.The Twiddle factor is defined as WN=e-j2 /N

15. Define Zero padding.The method of appending zero in the given sequence is called as Zero padding.

16. Define DFT.DFT is defined as X(w)= x(n)e-jwn.

Here x(n) is the discrete time sequence. X(w) is the fourier transform ofx(n).17. Find Z transform of x(n)={1,2,3,4}

Given x(n)= {1,2,3,4}X(z)= x(n)z-n

= 1+2z-1+3z-2+4z-3 = 1+2/z+3/z2+4/z3.18. Obtain the inverse z transform of X(z)=1/z-a,|z|>|a|

Given X(z)=z-1/1-az-1

By time shifting property, X(n)=an.u(n-1)

PART-B

1. What is ROC? State some properties of Z transform. (OCT / NOV 2002).2. Find the inverse Z transform of X(z) = [z+4] / [z2 – 4z + 3]. (OCT / NOV 2002).3. Find the Z transform of x(n) = an sin ω0n u(n). (OCT / NOV 2002). 4. How will you eveluate the fourier transform from pole zero plot of Z transform? (OCT / NOV 2002).

5. Find the inverse Z transform of X(z) = 1 / [1-1.5z-1 + 0.5z-2] for ROC : 0.5 < |z| < 1. 6. Write down any four properties of z transform and explain. (APR / MAY 2003).


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7. Find z transform of [u(n) – u(n-10)]. (APR / MAY 2003).8. Obtain the relation between z transform and DTFT. ( NOV/DEC 2003).9. Find the final value of the given signal W(z) = 1 / [1+2z-1-3z-2] and z transform of cos βnt. ( NOV/DEC 2003).

10. Find the inverse z transform using contour integral method. Given, X(z) = 1 / [1 – az-1], |z| > a. ( NOV/DEC 2003).

11. Find the IZT of X(z) = [1 – 1/3z-1] / [(1 – z-1)(1 + 2z-1)], |z| > 2.(MAY 2004).12. Find the DFT of x(n)={1,1,1,1,1,1,0,0}13. Find the circular convolution of x1(n)={1,2,0,1}& x2(n)={2,2,1,1}14. Find the z transform of the sequence x(n) = [1 / 2]n u(n) – [1 / 4]u(n - 1). (DEC 2004).15. What are the 3 possible sequences whose z transform is given by X(z) = [8/6

z2 – 67/12z] / [z2 – 17/12z + 1/2]. ( NOV/DEC 2004).16. Find the z transform and its ROC of x(n) = 1, n ≥ 0

= 3n , n < 0. (APR / MAY 2005).17. State and prove convolution property of z transform. (APR / MAY 2005).18. Find the DTFT of x(n) = 0.5n u(n) and plot its spectrum. (APR / MAY 2004).19. Determine the DTFT of x(n) = 1, for 0 ≤ n ≤ 5 & 0, otherwise. (APR / MAY 2005).

UNIT-V - LINEAR TIME INVARIANT DISCRETE TIME SYSTEMS

PART-A

1. Define convolution sum?If x(n) and h(n) are discrete variable functions, then its convolution sumy(n) is given by,y(n)=_ x(k) h(n-k)

2. List the steps involved in finding convolution sum?o foldingo Shiftingo Multiplicationo Summation

3. Define LTI causal system?A LTI system is causal if and only if ,h(n) = 0 for n<0.This is the sufficientand necessary condition for causality of the system.

4. Define LTI stable system? The bounded input x(n) produces bounded output y(n) in the LTI system only if, ∑| h(k)| <∞. When this condition is satisfied ,the system will be stable.5. If u(n) is the impulse response response of the system, What is its step response?

Here h(n) = u(n) and the input is x(n) = u(n).Hence the output y(n) = h(n) * x(n)

= u(n) * u(n)6. Convolve the two sequences x(n)={1,2,3} and h(n)={5,4,6,2}

y(n)={5,14,29,26,22,6}7. Is the output sequence of an LTI system is finite or infinite when the input x(n) is

finite? Justify your answer.(OCT / NOV 2002)If the impulse response of the system is infinite, then output sequence is infinite even though input is finite. For example, consider,


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Input x(n) = δ(n) finite length.Impulse response, h(n) = an u(n) infinite length.Output sequence, y(n) = h(n) * x(n)

= an u(n) * δ(n) = an u(n).8. Consider an LTI system with impulse response h(n) = δ(n – n0) for an input x(n),

find the Y(ejω). (NOV / DEC 2003).Here Y(ejω) is the spectrum of output. By convoution theorem we can write,

Y(ejω) = H(ejω) X(ejω)Here H(ejω) = DTFT {δ(n – n0)} = e-jωn0

Y(ejω) = e-jωn0X(ejω)9. Find the linear convoution of x(n) = {1, 2, 3, 4, 5,6} with y(n) = {2, -4, 6, -8}.

(APR / MAY 2004).y(n) = {2, 0, 4, 0, -4, -8, -26, -4, -48}.

10. Determine the system function of the discret time system described by the difference equation y(n) – 0.5y(n - 1) + 0.25y(n - 2) = x(n) – x(n - 1).(APR / MAY 2004).Taking z transform on both sides,

Y(z) – 0.5z-1Y(z) + 0.25z-2Y(z) = X(z) – z-1X(z).H(z) = Y(z) / X(z) = [1 – z-1] / [1 – 0.5z-1 + 0.25z-2].

11. What is the linear convolution of the two signals {2, 3, 4} and {1, -2, 1}?(NOV / DEC 2004).y(n) = {2, -1, 0, -5, 4}.

12. What is the response of an LSI system with impulse response h(n) = δ(n) + 2 δ(n - 1) for the input x(n) = {1, 2, 3}?(APR / MAY 2005).Here h(n) = δ(n) + 2 δ(n - 1) can be expressed as h(n) = {1, 2}y(n) = {1, 4, 7, 6}.

13. Write the general difference equation relating input and output of a system. (APR / MAY 2003).The generalized difference equation is given as

N My(n) = -∑ ak y(n - k) + ∑ bk x(n -k). k = 1 k = 0here y(n - k) are previous outputs and x(n - k) are present and previous inputs.

14. How unit sample response of discrete time system is defined? The unit step response of the discrete time system is output of the system tounit sample sequence. i.e., T[ð(n)]=h(n). Also h(n)=z {H(z)}.

15. State the maximum memory requirement of N point DFT including twiddle factors? [2N+N/2]

16.What is the impulse response x(n) of the system if the poles and zeros are radially moved K times their original location?(NOV / DEC 2004) If the impulse response is an u(n).then new impulse response after movement of poles by K times will be kn an u(n). i) For k<1,impulse response will decay fast. ii) For k>1,impulse response will decay very slowly.There is no effect on impulse response due to movement of zeros

PART-B1. State and prove the properties of convolution sum?2. Determine the convolution of x(n)={1,1,2} h(n)=u(n) graphically?3. Determine the forced response for the following system


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y(n)-1/4 y(n-1) – 1/8 y(n-2) = x(n) + x(n-1) for x(n)=(1/8)n u(n) . Assume zero initial conditions?4. Compute the response of the system y(n)=0.7 y(n-1) – 0.12 y(n-2) + x(n-1) – x(n-2) to the input x(n) = n u(n). Is the system is stable?5.Find the output of the system whose input – output is related by, y(n) = 7 y(n-1) –

12 y(n-2) + 2 x(n) –x(n-2) for the input x(n) = u(n). (OCT/NOV 2002)6.Find the linear convolution of x(n) = {1,2,3,4} and h(n) = {2,3,4,1}. (NOV 2002)7.Find the impulse response of the stable system whose input – output is relation is given by the equation y(n)- 4 y(n-1)+ 3 y(n-2) = x(n) + 2x(n-1). (APR / MAY 2003).8.Find the overall impulse response of the system shown in figure. (MAY 2003).

9.Given y(n) =x(n) + 1/8 x(n-1)+1/3 x(n-2).Find whether the system is stable or not. (NOV / DEC 2003)

10.Compute the convolution of the two sequences given and plot the output.(DEC 2003)

11.Find the output sequence y(n) of the system described by the equation y(n) = 0.7 y(n-1) – 0.1 y(n-2) + 2 x(n) – x(n-2).For the input sequence x(n) = u(n). (DEC 2004) 12. Find the overall impulse response of the causal system shown in figure. (DEC 2004)

Here h1(n) = (1/3)n u(n), h2(n) = (1/2)n

u(n) and h3(n) = (1/5)n u(n)

13.Find the convolution of x(n) = {1,2,3,4,5} with h(n) = {1,2,3,3,2,1}. (MAY 2005). 14.Find the impulse response of the discrete – time system described by the difference equation, y(n-2) – 3 y(n-1) + 2 y(n) = x(n-1). (APR / MAY 2005). 15.Realize direct form – I, direct form – II, cascade and parallel realization of the discrete time system having the system function

H(z) = [2(z+2)] / [z(z-0.1)( z+0.5)( z+0.4)] (APR / MAY 2004).KSRCE/ECE SIGNALS AND SYSTEMS EC403

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