IT1201S&sUnitNotes

8/20/2019 IT1201S&sUnitNotes

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SIGNALS & SYSTEMS

Unit II

Fourier series representation of continuous-time periodic signals:

Complex Fourier seri es expansion:The Fourier series expansion of a continuous-time periodic, x(t) signal is given by

∞

x(t)=Σ ck e jk ω0tk=−∞

or

∞ x(t)=Σ ck e j2πkt/T

k=−∞

where ck are Fourier series coefficients given by

T

ck =(1/T)∫ x(t) e− jk ω0t dt0

or

T

ck =(1/T)∫ x(t) e− j2πkt/T dt0

where, in all equations,

ω0 = (2π/T)

Convergence of F ouri er seri es: Di ri chlet Conditions: The Fourier series expansion of a periodic signal defined by the equation

∞

x(t)=Σ ck e jk ω0tk=−∞

is convergent i.e., the Fourier series expansion of a periodic signal is possible if and only if

the signal has finite energy over one period or the signal is square-integrable over one period

i.e.,


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T

∫ |x(t)|2 dt < ∞ 0

or if the signal satisfies the following conditions known as the Di ri chlet conditions .

Conditi on 1: x(t) must be absolutely integrable over one period i.e.,

T

∫ |x(t)| dt < ∞ 0

Conditi on 2: The signal has a finite no. of maxima and minima in any finite interval of time.

Conditi on 3: The signal has finite no. of finite discontinuities in any finite interval of time.

Tr igonometr ic Fourier series expansion:The Fourier series expansion of a continuous-time periodic, x(t) signal is given by

∞

x(t)=(a0/2) +Σ [ak cos(k ω0t) + bk sin(k ω0t)] k=1

where

T

ak =(2/T)∫ x(t) cos(k ω0t) dt0

T

bk =(2/T)∫ x(t) sin(k ω0t) dt0

T

a0=(2/T)∫ x(t) dt0

and (a0/2)=c0; ak =[ck + c−k ]; bk =j[ck − c−k ].

Even and odd signals:

A signal, x(t) is said to be even if x(t)=x(−t). For an even signal, bk =0 and hence thetrigonometric Fourier series becomes


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∞

x(t)=(a0/2) +Σ ak cos(k ω0t) k=1

where

T

ak =(2/T)∫ x(t) cos(k ω0t) dt0

T

a0=(2/T)∫ x(t) dt0

A signal, x(t) is said to be odd if x(t)= −x(−t). For an odd signal, ak =0 and hence thetrigonometric Fourier series becomes

∞

x(t)=Σ bk sin(k ω0t) k=1

where

T

bk =(2/T)∫ x(t) sin(k ω0t) dt0

Fourier transform of aperiodic continuous-time signals: Spectrum of aperiodic

continuous-time signals:

The Fourier transform of an aperiodic continuous-time signal, x(t) is defined as

∞

X(ω)=∫ x(t) e− jωt dt−∞

where X(ω) is also called the spectrum of x(t). Recovering the signal x(t) from X(ω) i.e., theinverse Fourier transform is defined as

∞

x(t)=(1/2π)∫ X(ω) e jωt dω −∞


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Properti es of continuous-time Fourier Transform:

1. Linearity: ax1(t) + bx2(t) ↔ aX1(ω) + bX2(ω)

2. Time shi fting: x(t−t0) ↔ e− jωt0 X(ω)

3.

Frequency shi f ting: e− jω0t x(t) ↔ X(ω−ω0)

4. Time scaling: x(at) ↔ (1/|a|)X(ω/a)

5. Time reversal: x(−t) ↔ X(−ω)

6. Duality: X(t) ↔ 2πx(−ω)

7. Di ff erentiation in time domain: [dx(t)/dt] ↔ jωX(ω)

8.

Di ff erentiation in f requency domain: (− jt)x(t) ↔ [dX(ω)/dω] t

9. I ntegration in time domain: ∫ x(t) dt ↔ πX(0) δ(ω) + (1/jω)X(ω) −∞

10. Convolution: x1(t)∗x2(t) ↔ X1(ω)X2(ω)

11. Multiplication: x1(t)x2(t) ↔ (1/2π)X1(ω)∗X2(ω) ∞ ∞

12. Parseval’ s theorem: ∫ |x(t)|2 dt ↔ (1/2π)∫ |X(ω)|2 dω −∞ −∞

Laplace transform:

The Laplace transform of a continuous-time signal x(t) is defined as

∞

X(s)=∫ x(t) e−st dt−∞

Properties of L aplace transform:

1.

Linearity: ax1(t) + bx2(t) ↔ aX1(s) + bX2(s)

2. Time shi fting: x(t−t0) ↔ e−st0 X(s)

3. Frequency shi f ting: e−s

0t x(t) ↔ X(s−s0)

4. Time scaling: x(at) ↔ (1/|a|)X(s/a)

5. Time reversal: x(−t) ↔ X(−s)

6.

Di ff erentiation in time domain: [dx(t)/dt] ↔ sX(s)


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7. Di ff erentiation in f requency domain: (−t)x(t) ↔ [dX(s)/ds] t

8. I ntegration in time domain: ∫ x(t) dt ↔ (1/s)X(s) −∞

9.

Convolution: x1(t)∗x2(t) ↔ X1(s)X2(s)

Laplace transform pairs

∞

1. f(t)⇔F(s)=∫ f(t)e−stdt0−

n

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

3. (−t)nf(t)⇔dn[F(s)]/dsn 4. eatf(t)⇔F(s−a)5. e−atf(t)⇔F(s+a)6. δ(t)⇔1

7.

dn[δ(t)]/dsn⇔sn

8. u(t)⇔1/s9. tn/n!⇔1/sn+1

10. e

at⇔1/(s−a)

11. e−at⇔1/(s+a)12.

sinbt⇔b/(s2+b2)

13. cosbt⇔s/(s2+b2)14.

sinhbt⇔b/(s2−b2)

15. coshbt⇔s/(s2−b2)

16. eatsinbt⇔b/[(s−a)2+b2]17. eatcosbt⇔(s−a)/[(s−a)2+b2]18. e−atsinbt⇔b/[(s+a)2+b2]19. e−atcosbt⇔(s+a)/[(s+a)2+b2]20.

eat

[tn/n!]⇔1/(s−a)n+1

21.

e−at

[tn/n!]⇔1/(s+a)n+1


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SIGNALS & SYSTEMS

Unit III

Differential equations describing continuous-time LTI systems:The continuous-time LTI systems are described by the linear constant coefficient

differential equations of the form

N M

Σ bk [dk y(t)/dtk ]=Σak [dk x(t)/dtk ]k=0 k=0

or

b0y(t) + b1[dy(t)/dt] + … + b N[d Ny(t)/dt N]= a0x(t) + a1[dx(t)/dt] + … + aM[d

Mx(t)/dtM]

or

b0y(t) + b1y1(t) + b2y

2(t) + … + b Ny

N(t)= a0x(t) + a1x

1(t) + a2x

2(t) + … + aMx

M(t)

where

yi(t)=[diy(t)/dti] – ith derivative of y(t)

xi(t)=[dix(t)/dti] – ith derivative of x(t)

Transfer function of continuous-time LTI systems described by given differentialequation: The transfer function of an LTI system described by the given differential equation of

the form

b0y(t) + b1[dy(t)/dt] + … + b N[d N

y(t)/dt N

]= a0x(t) + a1[dx(t)/dt] + … + aM[dM

x(t)/dtM

]

or

b0y(t) + b1y1(t) + b2y

2(t) + … + b Ny N(t)= a0x(t) + a1x

1(t) + a2x2(t) + … + aMx

M(t)

is defined as the ratio of the Laplace transform of its output to that of its input i.e.,

H(s)=[Y(s)/X(s)]

Impulse response of continuous-time LTI systems described by given differential

equation:

The impulse response of a continuous-time LTI system described by the givendifferential equation is the inverse Laplace transform of its transfer function i.e.,

h(t)=L −1[H(s)]

Convolution integral:The convolution integral is defined by


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∞

y(t)=x(t)∗h(t)=∫ x(τ)h(t−τ)dτ −∞


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SIGNALS & SYSTEMS

Unit IV

Discrete-time signals: A discrete-signal is a sequence of numbers x in which the nth number in the sequence is

denoted as

x(n), −∞≤n≤∞

i.e., a discrete-time signal is a function whose independent variable is the set of integers.

Some special discrete-time signals:

(1) Uni t-sample or impulse sequence:

Unit-sample or impulse sequence δ(n) is defined as

1 if n=0

δ(n)=0 if n≠0

The shifted unit-sample or impulse sequence δ(n−k) is defined as

1 if n= −k or +kδ(n±k)=

0 if n≠k

(2) Uni t-step sequence: Unit-step sequence u(n) is defined as

1 if n≥0

u(n)=0 if n


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The shifted unit-step sequence u(n−k) is defined as

1 if n≥−k or +ku(n±k)=

0 if n


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I nverse Discrete-Time Fourier Transform (IDTFT):

The inverse Fourier transform of X(ω) is defined as

ω0+2π

F−1

[X(

ω)]=x(n)=(1/2

π)

∫ X(

ω)e

jωnd

ω

ω0

where, in many cases, ω0=−π so that

Discrete Fouri er Transform (DFT): The discrete Fourier transform of a finite-length sequence x(n) is defined as

N−1

X(k)=Σx(n)e− j2π(n/N)k , 0≤k ≤ N−1n=0

or

N−1

X(k)=Σx(n)W Nnk , 0≤k ≤ N−1n=0

where W N= e− j(2π/N).

X(k) is periodic with period N i.e., X(k+N)=X(k).

I nverse Discrete Fourier Transform (IDFT):

The inverse discrete Fourier transform of X(k) is defined as

N−1

x(n)=(1/N)ΣX(k)e j2π(n/N)k , 0≤n≤ N−1k=0

or

N−1

x(n)=(1/N)ΣX(k) W N−nk , 0≤n≤ N−1k=0

where W N= e− j(2π/N).

Properties of DFT:

Peri odicity property: If X(k) is the N-point DFT of x(n), then


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X(k+N)=X(k)

L ineari ty property:If X1(k)=DFT[x1(n)] & X2(k)=DFT[x2(n)], then

DFT[a1x1(n)+a2x2(n)]=a1X1(k)+a2X2(k)

Convoluti on property:If X1(k)=DFT[x1(n)] & X2(k)=DFT[x2(n)], then

DFT[x1(n) N x2(n)]=X1(k)X2(k)

where N indicates N-point circular convolution.

Mul tipli cation property:

If X1(k)=DFT[x1(n)] & X2(k)=DFT[x2(n)], then

DFT[x1(n)x2(n)]=(1/N)[X1(k) N X2(k)]

where N indicates N-point circular convolution.

Z transforms:Definition: The (two-sided or bilateral) Z transform of a discrete time function x(n) is

defined as

∞

Z [x(n)]=X(Z)=Σ x(n)Z−n n=−∞

Region of convergence (ROC): For any given sequence, the set of values of Z for which the Z transform of the

sequence absolutely converges is called the region of convergence, which is abbreviated as

ROC. For the absolute convergence of the Z transform of a given sequence x(n), the

condition is

∞

Σ x(n)Z−n


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Z-tr ansform pairs

∞

1. x(n)⇔X(Z)=Σx(n)Z−n n= −∞

2. x(n−k)⇔Z−k X(Z)

3. anx(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.

anδ(n)⇔1

9. anδ(n−k)⇔Z−k ak 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.

anu(n−k)⇔Z−k ak [1/(1−aZ−1)]

16. e jnθu(n)⇔[1/(1−e jθZ−1)]17.

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

18. sinnθu(n)⇔[Z−1sinθ/(1−2Z−1cosθ+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

Classification of discrete-time systems:

L inear system:

A system is said to be linear if, for some linear combination of any number of inputs(excitations) x1(n), x2(n), x3(n), …, it produces same linear combination of individual outputs

(responses) y1(n), y2(n), y3(n), …, i.e.,

R [ax1(n) + bx2(n) + cx3(n) + …]=ay1(n) + by2(n) + cy3(n) + …

for all possible values of the constants a, b, c, ….

Time-invariant system:

A system is said to be time-invariant if the output (response) of the system to any

input (excitation) with a time shift or delay, is the same output (response) with the same time

shift or delay i.e.,

R [x(n−k)]=y(n−k)

Causal systems:

A system is said to be causal if the output (response) of the system does not depend

on future input (excitation) values or in other words if the output (response) of the systemdepends only on past or present input (excitation) values i.e., the system is ‘nonanticipate’.

Stable system – BI BO stabil i ty:

A system is said to be stable if it produces a bounded output (response) for a bounded

input (excitation) i.e.,

x(n)


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or by

N M

Σ bk y(n−k)=Σ

am x(n−m)k=0 m=0

The LTI systems described by the above equation are called the recursive systems

(i.e., systems with feedback). With bk =0 for k=1, 2, …, N, the above equation becomes

b0 y(n)=a0 x(n) + a1 x(n−1) + … + aM x(n−M)

or by

M

y(n)=Σ (am/b0) x(n−m)m=0

The LTI systems described by the above equation are called the non-recursive

systems (i.e., systems with feedback).

The impulse response (IR) of a system:

Definition:The impulse response, h(n) of a system is defined as the system response to the unit-

sample or impulse sequence δ(n).

Impulse response of a linear, time-invariant, causal system described by the standard

Nth order difference equation:

The impulse response of a LTI, causal system described by the standard Nth orderdifference equation

y(n)+b1y(n−1)+b2y(n−2)+…+b Ny(n− N)=a0x(n)+a1x(n−1)+…+aMx(n−M)(assuming zero initial conditions i.e., y(n)=0 for n


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i. If the roots obtained are distinct, say λ1, λ2,…, λ N, then the homogeneous solutionwould be

yh(n)=[C1(λ1)n+C2(λ2)n+…+C N(λ N)n]u(n).ii. If any of the roots is a multiple order root, say λ1 of order m, then the

homogeneous solution would be

yh(n)={[C1(λ1)n+C2n(λ1)n+C3n2(λ1)n+C4n3(λ1)n+…+Cmnm−1(λ1)n]

+Cm+1(λm+1)n+Cm+2(λm+2)n+…+C N(λ N)n}u(n).where C1, C2, C3… are arbitrary constants.

Procedure Block 2: Impulse Response h(n):

7) Form the impulse response h(n) as follows: Note down the maximum shift on y (i.e., N) and maximum shift on x (i.e., M).

i. If M


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∞

y(n)=R [ Σ x(k) δ(n−k)] ---------------------------------------------------------------------(3)k=−∞

Since k is a dummy variable and R operates only on n which indexes both the input

and output sequences, by the linearity of the system,

∞

y(n)=Σ x(k) R [δ(n−k)] --------------------------------------------------------------------------(4)k=−∞

If h(n) represents the response of the system to the unit-sample or impulse sequence,

δ(n) or as usually called, the impulse response of the system, then, by the time-invariance ofthe system,

∞

y(n)=Σ x(k) h(n−k) ------------------------------------------------------------------------------(5)k=−∞

The above equation is generally known as convolu tion sum and suggests the

characterization of a linear time-invariant system completely by its impulse response, in the

sense that, given h(n), it is possible to use Equ(5) to find the response of the system to any

excitation x(n) and to define the properties of the system.

Property 1: The convolution operation is commutative.

x(n) ∗ h(n) = h(n) ∗ x(n)

Property 2: The convolution operation is distributive over addition

x(n)*[h1(n)+h2(n)] = x(n)*h1(n) + x(n)*h2(n)

Convoluti on of two inf in i te-length sequences:

The convolution y(n) of two infinite-length sequences x(n) and h(n) is defined as

Rx(n) y(n)

Fig1


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∞

y(n) =Σx(k)h(n−k), −∞ ≤ n ≤ ∞ k=−∞

or

∞

y(n) =Σh(k)x(n−k), −∞ ≤ n ≤ ∞ k=−∞

Determinati on of stabil i ty of a LTI system, given its impulse response:

An LTI system is stable if its impulse response, h(n) is absolutely summable i.e,

∞

Σ h(n)


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Z-transform analysis of LTI systems:

Transfer function of an LT I system: The transfer function of an LTI system is defined as the ratio of the Z-transform of its

output to that of its input i.e.,

H(Z) = [Y(Z)/X(Z)]

Impul se response of an LTI system descri bed by the given dif ference equation: The impulse response of an LTI system is the inverse Z-transform of the transfer

function H(Z) i.e.,

h(n)=Z−1[H(Z)]

Frequency response of an LTI system:The frequency response of an LTI system is defined as the Z-transform computed at

the unit circle in the Z plane i.e.,

H(e jω)=H(Z)|Z=e jω

Response of an LTI system descri bed by given di ff erence equation to any input, x(n ):The response of an LTI system described by the given difference equation of the form

b0 y(n) + b1 y(n−1) + … + b N y(n− N)=a0 x(n) + a1 x(n−1) + … + aM x(n−M)

to any input x(n) is given by

y(n)=Z−1[H(Z)X(Z)]

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