1

Time and Frequency Characterization of Signals & Systems

•Frequency Domain Characterization through multiplication of FourierTransform of input signal and system frequency response.( Transfer Function).

•Time Domain Characterization through convolution of input signal and system impulse response.

Convenient to use frequency domain because easy operation of Multiplication as oppose to operation of convolution in time domain.

2

Magnitude and Phase Representation of Fourier Transforms

)()(

|)(|

.|)(|)(

)()(

|)(|

.|)(|)(

.

)(

)(

j

j

eXjjj

jXj

eXPhase

eXMagnitude

eeXeX

TransformFouriertimeDiscrete

jXPhase

jXMagnitude

ejXjX

TransformFouriertimeContinuous

j

3

)(X je)(e H )X(e)Y(e jjj

)(X)(H)(Y jjj

Magnitude-Phase Representation of The Frequency Response of LTI Systems

h(t)H(j

x(t) y(t)=h(t)*x(t)

X(j Y(jjX(j

h[n]

x[n] y[n]=x[n]*h[n]

|Y(jjX(j

)H(e j

4

)(X)(H)(Y jjj

Linear Phase and Group Delay of LTI Systems

h(t)H(j

x(t) y(t)=h(t)*x(t)

X(j Y(jjX(j

|Y(jjX(j

)}.({)(

).0

()(...0

)(,

.0

)(1|)(|..

.0)(,

jHd

dasdefineisdelayGroup

ttxtyeitbyinputthedelayshifttoisdoessystemthisWhat

tjHandjHei

tjejHifPhaseLinear

5

|)(|log20 10 H

Log-Magnitude and Bode plots

The absolute values of the magnitude of the transfer function of a system are normally converted into decibels define as

.

)(log)(

|)(|log20

10

10

plotsBodeasknownare

versusjHand

jHofPlots

6

Continuous-time Filters Described By Differential Equations.

Simple RC Lowpass Filter.

R

C

+

+ -

-vs(t)

vr(t)

vc(t)

RCjjH

eejHejHdt

dRC

eHtVc

etVs

tVstVcdt

tdVcRC

tjtjtj

tj

tj

1

1)(

)(])([

)()(

,)(

)()()(

dt

tdvcCti

)()(

dt

tdvcCti

)()(

h(t)H(

vs(t) Vc(t)=h(t)*vs(t)

Vs( Vc(Vs(

7

Continuous-time Filters Described By Differential Equations.

Simple RC Lowpass Filter.

R

C

+

+ -

-vs(t)

vr(t)

vc(t)

RCjjH

tvstvcdt

tdvcRC

1

1

)Vs(

)Vc()(

)Vs()Vc()Vc(RCj

equation. of sideboth on F.T. Taking

)()()(

Filter. theof

responsefrequency or function transfer

at the getting of method eAlternativ

8

First-order Recursive Discrete-time Filter

j-j

nj1)-n(jjnjj

njjnj

e1

1)H(e

.ee)H(ee)H(e

equation previous in the

.e)H(ey[n] then ,ex[n]

system LTI ofProperty Function Eigen From

][]1[][

a

a

ngSubstituti

nxnayny

9

First-order Recursive Discrete-time Filter

j-j

j-

e1

1)H(e

)(

)Y(

)()(ae-)Y(

-:equation above of DTFT Taking

][]1[][

aX

XY

nxnayny

D

+

a

x[n] y[n]

y[n-1]

ay[n-1]

h[n]H(

x[n] y[n]=x[n]*h[n]X( Y(X(

10

First-order Recursive Discrete-time Filter

j-

.

j.

e1

1][

)H(e][

anua

nhTF

n

TF

11

Impulse response of First order recursive D-T lowpass filter

10],[][ anuanh n

n0

jn

nj

n

njnj

aeae

enuaeH

1

1)(

][)(

0

12

Frequency Response of First order recursive lowpass filter

10],[][ anuanh n

jj

aeeH

1

1)(

)1(

1

a

)1(

1

a

2-

|H(

0Phase H(

2-

)1/(tan 21 aa

)1/(tan 21 aa

13

Impulse-Train Sampling

n

nTttp )()(

n

p nTtnTxtptxtx )()()()()(

14

Multiplication/Modulation Property

x(t)

X(

djPjXtptxtx

TF

p )(()(2

1)()()(

.

X

p(t)

P(

ksp

ks

p

kjXT

jX

jXjX

kT

jP

djPjXjwX

)(1

)(

))(()(*)(

)(2

)(

)(()(2

1)(

00

)](*)([2

1)(

)()()(

jPjXjX

tptxtx

p

p

15

Convolution in Frequency Domain

16

Sampling Theorem

./..

./2)(2

;cov)(

.||0)(..lim)(

,....2,1,0

][sec)( sec

M

sMs

M

T

TperiodSamplingei

TwhererateNyquist

ratetheatsampleweiferedreuniquelybecantx

forXeiitedbandistx

n

nTXTeverysampledistx

17

Violating Sampling Theorem resulting in aliasing.

18

Reconstruction using an ideallowpass filter.

19

Reconstruction looking from the time domain-convolving.

h(t)

H(j Y(jjX(j

)(txp )(*)()( thtxty p

)(}.)()(

).(}.)()({)(

)(*})()({)(

)(*)()(

)()(

x(t).y(t),)π

tωSinc(

π

ωTh(t))()(

frequency)(cutoffωωωfor1,H(ω(filterlowpassidealanFor)()()(

cc

cc

nTthnTxty

nTuttingp

dthnTxty

thnTttxty

thtxty

nTttx

nTttx

tptxtx

n

n

n

p

n

n

p

x

p(t)

x(t) )( jX p

20

Continuous to discrete-time signal conversion.

21

Discrete-time Processingof Continuous-time Signals.

22

Reconstruction of a sampled signal with a zero-order hold

23

Comparison of Frequency Responses (Transfer Functions) of ideal lowpass reconstruction filter and zero-order hold

reconstruction filter.

24

Reconstruction of a sampled signal with a first-order hold

25

Comparison of Frequency Responses (Transfer Functions) of ideal lowpass reconstruction filter, zero-order hold

reconstruction filter and first-order hold reconstruction filter.

26

Reconstruction of a sampled signal with ideal lowpass filter