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SIGNALS & SYSTEMS

Unit II

1. Define complex Fourier series expansion of continuous-time periodic signals.

2. What are the Dirichlet conditions?

3.

Expand the following signals in complex Fourier series.(i)

x(t)=sinω0t(ii)

x(t)=1+sinω0t+2cosω0t+cos[2ω0t+(π/4)]

1, |t|

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T/2− T/2

A1

−A2

Fig(c)

x(t)

t

A

Fig(d)

T1

− T1

−4T1 −2T1 4T12T1

x(t)

t

T− T

A

Fig(e)

x(t)

t0

T− T

A

Fig(f)

x(t)

t0

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T− T

A

Fig(g)

x(t)

t0

[Hint: Fig(a): x(t)=[2A/T]t, −T/2

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T1

− T2

A

Fig(b)

x(t)

t

T/2− T/2A1

−A2

Fig(c)

x(t)

t

A

Fig(d)

T1− T1−4T1 −2T1 4T12T1

x(t)

t

T− T

A

Fig(e)

x(t)

t0

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T− T

A

Fig(f)

x(t)

t0

T− T

A

Fig(g)

x(t)

t0

[Hint: Fig(a): x(t)=[2A/T]t, −T/2

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7. Compute the inverse Fourier transform of the following aperiodic continuous-time

signals

1

(i) X(ω)= ------------[(a+jω)]

(ii) H(ω)=jω (iii)

X(ω)=2πδ(ω−ω0)

∞

(iv) X(ω)=(2π/T)Σ δ(ω−k ω0) k= −∞

1, |ω|W

2sinωT1(vi)

X(ω)= -----------ω

(vii) X(ω)=1(viii) X(ω)=2πδ(ω)+πδ(ω−4π)+2πδ(ω+4π)

2, 0≤ω≤2(ix)

X(ω)= −2, −2≤ω2

(x) Y(ω)=[1/(a+jω)(b+jω)] for b≠a & b=a.8. Verify the Linearity, Time shifting, Frequency shifting, Time scaling, Time reversal,

Duality, Differentiation in time domain, Differentiation in frequency domain,

Integration in time domain, Convolution, Multiplication & Parseval’s theorem for the

CT Fourier transform.

9. Compute Fourier transform of x(t)=sinbt+cosbt. Use linearity property.

10. Compute Fourier transform of x(t)=δ(t)−e−2tu(t). Use linearity & frequency shifting

property.11. Compute the spectrum of the output of the system with impulse response h(t)=e−atu(t) to

the input x(t)=e− btu(t). [Hint: Find H(ω)X(ω)]12. Compute the spectrum of the output of the system with impulse response h(t)=e−atu(t) to

the input x(t)=u(t). [Hint: Find H(ω)X(ω)]13.

Compute the Laplace transforms of the following signals.

n

(i) dn[f(t)]/dt⇔sF(s)−Σ sn−i f i−1(0−)

i=1

(ii) (−t)nf(t)⇔dn[F(s)]/dsn (iii)

eatf(t)⇔F(s−a)(iv) e−atf(t)⇔F(s+a)

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(v) δ(t)⇔1(vi) dn[δ(t)]/dsn⇔sn (vii) u(t)⇔1/s(viii) tn/n!⇔1/sn+1 (ix) eat⇔1/(s−a)(x)

e−at⇔1/(s+a)(xi) sinbt⇔ b/(s2+b2)(xii) cosbt⇔s/(s2+b2)(xiii) sinhbt⇔ b/(s2− b2)(xiv) coshbt⇔s/(s2− b2)(xv)

eatsinbt⇔ b/[(s−a)2+b2](xvi)

eatcosbt⇔(s−a)/[(s−a)2+b2](xvii)

e−atsinbt⇔ b/[(s+a)2+b2](xviii)

e−atcosbt⇔(s+a)/[(s+a)2+b2](xix) eat[tn/n!]⇔1/(s−a)n+1 (xx)

e−at[tn/n!]⇔1/(s+a)n+1 14. Prove the Linearity, Time shifting, Frequency shifting, Time scaling, Time reversal,

Differentiation in time domain, Differentiation in frequency domain, Integration in time

domain & Convolution for the Laplace transform.

15.

Compute the inverse Laplace transforms of the following functions.

(i)

X(s)=[s+1]/[s2−1](ii)

X(s)=[s3−1]/[s2+s+1](iii) H(s)=[1/(s+2)]

(iv) H(s)=[1/(s2+16)]

(v) H(s)={1/[(s+0.1)2+9]}

(vi) H(s)={(s+0.2)/[(s+0.2)2+9]}

(vii)

H(s)={1/[(s+1)(s+2)]}

(viii) H(s)={1/[(s+0.5)(s2+0.5s+2)]}

(ix)

H(s)={2/[(s+1)(s+2)]}

(x)

H(s)=[Ωc/(s+Ωc)](xi) H(s)=[1/(s+1)2]

(xii) H(s)={(2s2+3s+3)/[(s+1)(s

2+2s+2)]}

(xiii)

H(s)=[2s/(s2+0.2s+1)]

(xiv) H(s)={s3/[(s+1)(s2+s+1)]}

(xv) H(s)=[Ωc2/(s

2+bk Ωcs+Ωc

2)]

(xvi)

H(s)=[1/(s2+(2)1/2s+1)]

(xvii)

H(s)={(3s2

+4)/[(s+2)(s2

+0.5s+1)]}(xviii) X(s)=[s

2+2s+5]/[s+3][s+5]

2

(xix) X(s)=[s3+2s2+6]/[s2+3s]

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SIGNALS & SYSTEMS

Unit III

1. Compute the transfer function of the following LTI systems.

(i) [dy(t)/dt]+3y(t)=x(t)

(ii)

RC[dy(t)/dt]+LC[d2y(t)/dt2]+y(t)=x(t)(iii) [d2y(t)/dt2]+3[dy(t)/dt]+2y(t)= [dx(t)/dt]+3x(t)

(iv) [dy(t)/dt]+(1/RC)y(t)=(1/RC)x(t)(v)

y/(t)+2y(t)=x(t)+x/(t)

(vi) y//(t)+y/(t)−2y(t)=x(t)2. Suppose that if the input to an LTI system is x(t)=u(t), then the output is y(t)=2e−3tu(t).

Compute its transfer function.

3. Suppose that if the input to an LTI system is x(t)=e−3t

u(t), then the output is

y(t)=[e−t−e−2t]u(t). Compute its transfer function.

4. Compute the impulse response of the following LTI system.

(i)

[dy(t)/dt]+3y(t)=x(t)(ii)

RC[dy(t)/dt]+LC[d2y(t)/dt2]+y(t)=x(t)

(iii) [d2y(t)/dt2]+3[dy(t)/dt]+2y(t)= [dx(t)/dt]+3x(t)

(iv) [dy(t)/dt]+(1/RC)y(t)=(1/RC)x(t)

(v) y/(t)+2y(t)=x(t)+x/(t)

(vi) y//(t)+y/(t)−2y(t)=x(t)5. Suppose that if the input to an LTI system is x(t)=u(t), then the output is y(t)=2e−3tu(t).

Compute its impulse response.

6. Suppose that if the input to an LTI system is x(t)=e−3t

u(t), then the output is

y(t)=[e−t−e−2t]u(t). Compute its impulse response.

7. Compute the impulse response of the following LTI system

[d3y(t)/dt3]+6[d2y(t)/dt2]+11[dy(t)/dt]+6y(t)=x(t)8. Compute the output of the LTI system with impulse response h(t)=e

−atu(t) to the

following input x(t)=|t|e−2|t|.

9. Compute the unit step response of the LTI system

[d2y(t)/dt

2]+3[dy(t)/dt]+2y(t)= [dx(t)/dt]+3x(t).

10.

Compute the convolution of the following signals.

(i) x(t)=e−atu(t) & h(t)=u(t)

(ii) x(t)=e2tu(−t) & h(t)=u(t−3)(iii) x(t)=t+1, 0≤t≤1, 2−t, 1

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SIGNALS & SYSTEMS

Unit IV

1. Define unit sample, unit step, ramp & exponential sequences.

2. Express the following sequence in terms of unit sample sequence x(n)={1,−1,1,1−1},

−2 ≤ n ≤ 2.3. Express the following sequence in terms of unit sample sequence

x(n)={1,1,−1,−1,−1,1,−1,1,−1,−1,1,1}.

4. Express the unit sample sequence in terms of unit step sequence.

5. Express the following sequence in terms of unit sample sequence x(n) = (1/2)n u(n).

6.

Express the following sequence in terms of unit sample sequence x(n) = u(n) − u(n− N)

7. Define Discrete-Time Fourier Transform (DTFT) of a sequence.8. Compute the Fourier transforms of the following sequences.

{1,2,0,−2,−1}, 0 ≤ n ≤ 4(i) x(n)=0, otherwise

(ii) x(n)=(1/2)nu(n)

(iii) x(n)=(1/2)nu(n-5)

(iv)

x(n)=δ(n)

1, 0 ≤ n ≤ M(v) wr (n)=

0, otherwise

(vi) x(n)=u(n)−u(n−6)(vii)

x(n)=2nu(−n)(viii) x(n)={−2,−1,0,1,2}

9. Find x(n) if

e− jωτ, |ω| ≤ ωc

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∞

1. x(n)⇔X(Z)=Σx(n)Z−n

n= −∞

2.

x(n−k)⇔Z−k

X(Z)

3. a

nx(n)⇔X(a

−1Z)=X(Z/a)

4. anx(n−k)⇔Z−k ak X(a−1Z)= Z−k ak X(Z/a)5.

ax1(n)+bx2(n)⇔aX1(Z)+bX2(Z)

6. δ(n)⇔17. δ(n−k)⇔Z−k

8. a

nδ(n)⇔1

9.

anδ(n−k)⇔Z

−k a

k

10. K δ(n)⇔K11. K δ(n−k)⇔KZ−k 12. u(n)⇔[1/(1−Z−1)]13.

u(n−k)⇔Z−k

[1/(1−Z−1

)]

14. anu(n)⇔[1/(1−aZ−1)]

15.

a

n

u(n−k)⇔Z

−k

a

k

[1/(1−aZ

−1

)]16.

e jnθ

u(n)⇔[1/(1−e jθ

Z−1

)]

17. cosnθu(n)⇔[(1−Z−1cosθ)/(1−2Z−1cosθ+Z−2)]18.

sinnθu(n)⇔[Z−1

sinθ/(1−2Z−1

cosθ+Z−2

)]

19. ancosnθu(n)⇔[(1−aZ−1cosθ)/(1−2aZ−1cosθ+a2Z−2)]20. ansinnθu(n)⇔[aZ−1sinθ/(1−2aZ−1cosθ+a2Z−2)]21.

nx(n)⇔Z−1

[dX(Z)/d(Z−1

)]

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SIGNALS & SYSTEMS

Unit V

1. Define linear systems.

2. Define time-invariant systems.

3.

Define causal systems.4. Define stable systems.

5. Define LTI systems.6.

Check the following systems for (a) linearity, (b) time-invariance and (c) causality.

(i) y(n)=ex(n)

(ii) y(n)=ax(n)+b

(iii) y(n)=anx(n)+b

(iv) y(n)=g(n)x(n)

(v) y(n)= x(n-m)

(vi) y(n)= a[x(n)]2+bx(n)

(vii)

y(n)=n[x(n)]2

(viii)

y(n)=x(n2)

(ix) y(n)=xe(n), the even part of x(n)(x)

y(n)=xo(n), the odd part of x(n)

(xi) y(n)=x(n) (xii) y(n)=cos[x(n)]

(xiii) y(n)=x(n)cosωon(xiv)

y(n)=x(-n+2)

(xv) y(n)=x(n)+nx(n+1)

(xvi)

y(n)=x(2n)

(xvii) y(n)=x(-n)

(xviii)

y(n)=x(n2

)7.

Identify whether the following systems are recursive or non-recursive systems.

(i) y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-y(n-1)-y(n-2)=x(n-2)

(iii) y(n)+y(n-1)=x(n)-2x(n-1)

(iv) y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

(v)

y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(vi) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)

(vii) y(n)=x(n)+2x(n-1)+x(n-2)(viii) y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(ix) y(n)=x(n)-2x(n-1)+x(n-2)

(x)

y(n)=x(n)+2x(n-1)-x(n-2)8. Define the impulse response of a system.

9. Determine the impulse response of the following LTI, causal systems.(i)

y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-(1/2)y(n-1)=x(n)+x(n-1)

(iii) y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)

(v) y(n)-2cosθy(n-1)+y(n-2)=sinθx(n-1)(vi)

y(n)+y(n-1)=x(n)-x(n-1)

(vii) y(n)+3y(n-1)+2y(n-2)=x(n)

(viii) y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)(ix)

3y(n)-y(n-1)=-x(n)+3x(n-1)

(x) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

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(xi) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)

(xii) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)

(xiii)

y(n)-2cosθy(n-1)+y(n-2)=x(n)(xiv)

y(n)-2cosθy(n-1)+y(n-2)=x(n)-cosθx(n-1)(xv) y(n)-3y(n-1)+2y(n-2)=x(n)+3x(n-1)+2x(n-2)

(xvi)

y(n)-2y(n-1)=x(n)+x(n-1)(xvii)

y(n)-ay(n-1)=x(n)+x(n-1)

(xviii) y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)(xix)

y(n)=0.6y(n-1)-0.08y(n-2)+x(n)

(xx) y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)

(xxi) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)

(xxii) y(n)+(1/2)y(n-1)=x(n)+2x(n-1)+x(n-2)

(xxiii) y(n)-y(n-1)=x(n)+x(n-1)

(xxiv)

y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)

(xxv) y(n)-5y(n-1)+6y(n-2)=2x(n-1)

(xxvi) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)

(xxvii)

y(n)=x(n)+2x(n-1)+x(n-2)(xxviii)y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(xxix) y(n)=x(n)-2x(n-1)+x(n-2)

(xxx) y(n)=x(n)+2x(n-1)-x(n-2)

10. What is the output of a system with the impulse response h(n) to the input x(n)?

11. Compute the convolution of the following sequences x(n)=an u(n) & h(n)=bn u(n).

12. Find the output of the system with impulse response h(n)=(1/2)n u(n) to the input

x(n)=1, 0≤n≤ N−1.13.

Compute the convolution of the following sequences x(n)=an u(n) &

h(n)=bn u(−n).

14.

Find the output of the system with impulse response h(n)=(1/2)

n

u(n) to the inputx(n)=(3/4)n u(n).

15. Determine the output of the system with impulse response h(n)={1,2,3} to the input

x(n)={1,−1,−1,1,1}.16. Determine the output of the system with impulse response

1, 0 ≤ n ≤ 6h(n)=

0, otherwiseto the input

(−1)n, 0 ≤ n ≤ 4x(n)=

0, otherwise17.

Find the output of the system with impulse response h(n)={2,1,−1} to the inputx(n)={−1,1,−1,−1,−1,1}.

18. Find the output of the system with impulse response h(n)={2,1,−2} to the inputx(n)={−1,2,1,1,−1,−1,1}.

19. Determine the output of the system with impulse response h(n)={1,2,3} to the input

x(n)={1,−1,−1,1,1}.20.

Determine the output of the system with impulse response

1, 0 ≤ n ≤ 6

h(n)= 0, otherwise

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to the input

(−1)n, 0 ≤ n ≤ 4x(n)=

0, otherwise

21. Find the output of the system with impulse response h(n)={2,1,−1} to the input

x(n)={−1,1,−1,−1,−1,1}.22. Find the output of the system with impulse response h(n)={2,1,−2} to the input

x(n)={−1,2,1,1,−1,−1,1}.23.

What is the condition on h(n) for a system to be stable?

24. Determine the impulse response of the following LTI, causal systems & hence its

stability.(i) y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-(1/2)y(n-1)=x(n)+x(n-1)(iii)

y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)


y(n)+y(n-1)=x(n)-x(n-1)

(vii) y(n)+3y(n-1)+2y(n-2)=x(n)

(viii) y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

(ix)

3y(n)-y(n-1)=-x(n)+3x(n-1)

(x) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(xi) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)(xii) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)

(xiii) y(n)-2cosθy(n-1)+y(n-2)=x(n)(xiv) y(n)-2cosθy(n-1)+y(n-2)=x(n)-cosθx(n-1)(xv) y(n)-3y(n-1)+2y(n-2)=x(n)+3x(n-1)+2x(n-2)

(xvi)

y(n)-2y(n-1)=x(n)+x(n-1)(xvii) y(n)-ay(n-1)=x(n)+x(n-1)

(xviii)

y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(xix) y(n)=0.6y(n-1)-0.08y(n-2)+x(n)

(xx) y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)(xxi) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)

(xxii) y(n)+(1/2)y(n-1)=x(n)+2x(n-1)+x(n-2)(xxiii)

y(n)-y(n-1)=x(n)+x(n-1)

(xxiv) y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)

(xxv) y(n)-5y(n-1)+6y(n-2)=2x(n-1)


(xxvii)


(xxix) y(n)=x(n)-2x(n-1)+x(n-2)

(xxx) y(n)=x(n)+2x(n-1)-x(n-2)25.

Determine the frequency response of the following LTI, causal systems.

(i) y(n)-(1/2)y(n-1)=x(n)(ii)

y(n)-(1/2)y(n-1)=x(n)+x(n-1)

(iii) y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)

(v) y(n)+y(n-1)=x(n)-x(n-1)

(vi) y(n)+3y(n-1)+2y(n-2)=x(n)

(vii)

y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

(viii) 3y(n)-y(n-1)=-x(n)+3x(n-1)

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(ix) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(x) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)

(xi) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)

(xii) y(n)-3y(n-1)+2y(n-2)=x(n)+3x(n-1)+2x(n-2)

(xiii)

y(n)-2y(n-1)=x(n)+x(n-1)

(xiv)

y(n)-ay(n-1)=x(n)+x(n-1)(xv) y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)(xvi) y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)

(xvii) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)(xviii)

y(n)-y(n-1)=x(n)+x(n-1)

(xix) y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)

(xx) y(n)-5y(n-1)+6y(n-2)=2x(n-1)

(xxi) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)

(xxii) y(n)=x(n)+2x(n-1)+x(n-2)

(xxiii)

y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(xxiv) y(n)=x(n)-2x(n-1)+x(n-2)

(xxv)

y(n)=x(n)+2x(n-1)-x(n-2)26. Determine the transfer functions of the following LTI, casual systems.

(i) y(n)-(1/2)y(n-1)=x(n)(ii)

y(n)-(1/2)y(n-1)=x(n)+x(n-1)

(iii) y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)

(v)

y(n)-2cosθy(n-1)+y(n-2)=sinθx(n-1)(vi) y(n)+y(n-1)=x(n)-x(n-1)

(vii) y(n)+3y(n-1)+2y(n-2)=x(n)(viii)

y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

(ix) 3y(n)-y(n-1)=-x(n)+3x(n-1)

(x)

y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(xi) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)

(xii) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)


(xvi) y(n)-2y(n-1)=x(n)+x(n-1)(xvii)

y(n)-ay(n-1)=x(n)+x(n-1)

(xviii) y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(xix) y(n)=0.6y(n-1)-0.08y(n-2)+x(n)

(xx)

y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)(xxi) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)

(xxii)

y(n)+(1/2)y(n-1)=x(n)+2x(n-1)+x(n-2)


(xxiv) y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)(xxv) y(n)-5y(n-1)+6y(n-2)=2x(n-1)

(xxvi) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)(xxvii)

y(n)=x(n)+2x(n-1)+x(n-2)

(xxviii)y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(xxix) y(n)=x(n)-2x(n-1)+x(n-2)

(xxx) y(n)=x(n)+2x(n-1)-x(n-2)

27.

Determine the transfer functions and hence the impulse responses of the following LTI,casual systems.

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(i) y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-(1/2)y(n-1)=x(n)+x(n-1)

(iii) y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)

(v) y(n)-2cosθy(n-1)+y(n-2)=sinθx(n-1)

(vi)

y(n)+y(n-1)=x(n)-x(n-1)(vii) y(n)+3y(n-1)+2y(n-2)=x(n)


(ix) 3y(n)-y(n-1)=-x(n)+3x(n-1)

(x) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(xi) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)

(xii) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)

(xiii) y(n)-2cosθy(n-1)+y(n-2)=x(n)(xiv) y(n)-2cosθy(n-1)+y(n-2)=x(n)-cosθx(n-1)(xv)

y(n)-3y(n-1)+2y(n-2)=x(n)+3x(n-1)+2x(n-2)

(xvi) y(n)-2y(n-1)=x(n)+x(n-1)

(xvii)

y(n)-ay(n-1)=x(n)+x(n-1)

(xviii) y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(xix) y(n)=0.6y(n-1)-0.08y(n-2)+x(n)

(xx)

y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)

(xxi) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)

(xxii)

y(n)+(1/2)y(n-1)=x(n)+2x(n-1)+x(n-2)


(xxiv) y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)(xxv)

y(n)-5y(n-1)+6y(n-2)=2x(n-1)


(xxvii)


(xxix) y(n)=x(n)-2x(n-1)+x(n-2)

(xxx)

y(n)=x(n)+2x(n-1)-x(n-2)

28. Determine the frequency response of the following LTI, causal systems using Z-

transform.(i) y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-(1/2)y(n-1)=x(n)+x(n-1)(iii)

y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)

(v) y(n)+y(n-1)=x(n)-x(n-1)

(vi)

y(n)+3y(n-1)+2y(n-2)=x(n)(vii) y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

(viii)

3y(n)-y(n-1)=-x(n)+3x(n-1)

(ix) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(x) y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)(xi) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)

(xii) y(n)-3y(n-1)+2y(n-2)=x(n)+3x(n-1)+2x(n-2)(xiii)

y(n)-2y(n-1)=x(n)+x(n-1)

(xiv) y(n)-ay(n-1)=x(n)+x(n-1)

(xv) y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(xvi) y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)

(xvii)

y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)(xviii)

y(n)-y(n-1)=x(n)+x(n-1)

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(xix) y(n)-y(n-1)+(1/4)y(n-2)=2x(n-1)

(xx) y(n)-5y(n-1)+6y(n-2)=2x(n-1)

(xxi) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)

(xxii) y(n)=x(n)+2x(n-1)+x(n-2)

(xxiii)

y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(xxiv)

y(n)=x(n)-2x(n-1)+x(n-2)(xxv) y(n)=x(n)+2x(n-1)-x(n-2)

29. Determine the forced response (output) of the following LTI, casual system to the

excitations (inputs)y(n)+3y(n-1)+2y(n-2)=x(n)

x(n)=4

x(n)=4(1/2)nu(n)

x(n)=(-1)nu(n)

x(n)=n(1/2)nu(n)

x(n)=n(-1)nu(n)

30. Determine the response (output) of the following LTI, casual system to the excitation

(input) x(n)=n2nu(n). y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)31. Determine the response (output) of the following LTI, casual system to the input

x(n)=n2nu(n). y(n)+4y(n-1)+4y(n-2)=x(n)+4x(n-1)+3x(n-2)

32.

Determine the response (output) of the LTI, casual system y(n)+2y(n-1)+y(n-2)=x(n) to

the input x(n)=9(2)n-8n+4.

33. Find the step responses of the following LTI, causal systems.

(i) y(n)-(1/2)y(n-1)=x(n)

(ii) y(n)-(1/2)y(n-1)=x(n)+x(n-1)

(iii)

y(n)-y(n-1)-y(n-2)=x(n-2)

(iv) y(n)+y(n-1)=x(n)-2x(n-1)


y(n)+y(n-1)=x(n)-x(n-1)

(vii) y(n)+3y(n-1)+2y(n-2)=x(n)


(ix)

3y(n)-y(n-1)=-x(n)+3x(n-1)

(x) y(n)-(1/4)y(n-1)-(3/8)y(n-2)=x(n)+x(n-1)

(xi)

y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)-x(n-1)

(xii) y(n)-(5/2)y(n-1)+y(n-2)=x(n)-x(n-1)


(xvi)

y(n)-2y(n-1)=x(n)+x(n-1)(xvii) y(n)-ay(n-1)=x(n)+x(n-1)

(xviii)

y(n)=(5/6)y(n-1)-(1/6)y(n-2)+x(n)

(xix) y(n)=0.7y(n-1)-0.1y(n-2)+2x(n)-x(n-2)

(xx) y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)(xxi) y(n)+(1/2)y(n-1)=x(n)+2x(n-1)+x(n-2)

(xxii) y(n)-y(n-1)=x(n)+x(n-1)(xxiii)

y(n)-5y(n-1)+6y(n-2)=2x(n-1)

(xxiv) 10y(n)-5y(n-1)+y(n-2)=x(n)-5x(n-1)+10x(n-2)

(xxv) y(n)=x(n)+2x(n-1)+x(n-2)

(xxvi) y(n)=(1/2)x(n)+x(n-1)+(1/2)x(n-2)

(xxvii)

y(n)=x(n)-2x(n-1)+x(n-2)(xxviii)y(n)=x(n)+2x(n-1)-x(n-2)

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