ECE 8443 – Pattern RecognitionECE 3163 – Signals and Systems

• Objectives:Response to a Sinusoidal InputFrequency Analysis of an RC CircuitResponse to Periodic InputsResponse to Nonperiodic InputsAnalysis of Ideal Filters

• Resources:Wiki: The RC CircuitCN: Response of an RC CircuitCNX: Ideal Filters

• URL: .../publications/courses/ece_3163/lectures/current/lecture_18.ppt• MP3: .../publications/courses/ece_3163/lectures/current/lecture_18.mp3

LECTURE 18: FOURIER ANALYSIS OF CT SYSTEMS

ECE 3163: Lecture 18, Slide 2

Response of an LTI System to a Sinusoid• Consider an LTI CT system with impulse response h(t):

• We will assume that the Fourier transform of h(t) exists:

• The output can be computed using our Fourier transform properties:

• Suppose the input is a sinusoid:

• Using properties of the Fourier transform, we can compute the output:)cos()( 0 tAtx

dtxhtxthty )()()(*)()(

dtethH tj )()(

)()()()()()(and)()()( XHYXHYXHY

)(cos)()()(

)(

)(

)()(

)(

)()()(

)(

0001

0)(

0)(

0

0)(

0)(

0

0000

00

00

00

00

HtHAYty

eeHA

eeeeHA

eHeHA

eeHA

XHY

eeAX

-

HjHj

jHjjHj

jj

jj

jj

F

ECE 3163: Lecture 18, Slide 3

• Using our FT properties:

Example: RC Circuit

)(1

)(1)(

txRC

tyRCdt

tdy

RCH

RC

RCH

RCj

RC

X

YH

XRCj

RCY

XRC

YRC

Yj

1

22

tan)(

)/1(

/1)(

/1

/1)()(

/1

/1)(

)(11

• Compute the frequency response:RC = 0.001;W=0:50:5000;H=(1/RC)./(j*w+1/RC);magH=abs(H);angH=180*angle(H)/pi;

ECE 3163: Lecture 18, Slide 4

Example: RC Circuit (Cont.)• We can compute the output for RC=0.001 and 0=1000 rad/sec:

• We can compute the output for RC=0.001 and 0=3000 rad/sec:

• Hence the circuit acts as a lowpass filter. Note the phase is not linear.

• If the input was the sum of two sinewaves:

describe the output.

451000cos)707.0()( tAty

6.713000cos)316.0()( tAty

tttx 3000cos100cos)(

ECE 3163: Lecture 18, Slide 5

Response To Periodic Inputs• We can extend our example to all periodic signals using the Fourier series:

• The output of an LTI system is:

• We can write the Fourier series for the output as:

• It is important to observe that since the spectrum of a periodic signal is a line spectrum, the output spectrum is simply a weighted version of the input, where the weights are found by sampling of the frequency response of the LTI system at multiples of the fundamental frequency, 0.

series)Fourier tric trigonome theof variant (acos)(1

00

k

kk tkAatx

1

0000 cos0)(k

kk kHtkkHAHaty

)(cand)(2

1c

also,

)()()0(a

where,

cos)(

0yk0

yk

000y0

100

kHkHA

kHkHAAHa

tkAaty

xk

xk

xk

yk

xk

yk

x

k

yk

yk

y

ECE 3163: Lecture 18, Slide 6

Example: Rectangular Pulse Train and an RC Circuit• Recall the Fourier series for

a periodic rectangular pulse:

• Also recall the system response was:

• The output can be easily written as:

)2/(

2/sin5.0

where,

cos)(

0

10

k

kaa

tkaatx

k

kk

RCj

RCH

/1

/1)(

even0

odd)/1()(

/12

)/1()(

/1

)2/(

2/sin)(

5.0)0(

where,

cos)(

22

220

0y0

100

k

kRCk

RC

k

RCk

RC

k

kkHAA

Haa

tkAaty

xk

yk

x

k

yk

yk

y

ECE 3163: Lecture 18, Slide 7

Example: Rectangular Pulse Train (Cont.)• We can write a similar expression for the output:

oddkk

y RCktkRCk

RC

katy

1

1

220 tancos)/1()(

/12)(

1/RC = 1

1/RC = 10

1/RC = 100

• We can observe the implications of lowpass filtering this signal.

• What aspects of the input signal give rise to high frequency components?

• What are the implications of increasing 1/RC in the circuit?

• Why are the pulses increasingly rounded for lower values of 1/RC?

• What causes the oscillations in the signal as 1/RC is increased?

ECE 3163: Lecture 18, Slide 8

Response to Nonperiodic Inputs• We can recover the output in the time domain using the inverse transform:

• These integrals are often hard to compute, so we try to circumvent them using transform tables and combinations of transform properties.

• Consider the response of our RC circuit to a single pulse:

• MATLAB code for the frequency response:RC=1;w=-40:.3:40;X=2*sin(w/2)./w;H=(1/RC)./(j*w+1/RC);Y=X.*H;magY=abs(Y);

deeXeHty tjjj )()(2

1)(

RCj

RC

eHeXeY

RCj

RCeH

eX

jjj

j

j

/1

/1

)2/(

)2/sin(

)()()(

/1

/1)(

)2/(

)2/sin()(

ECE 3163: Lecture 18, Slide 9

Response to Nonperiodic Inputs (Cont.)• We can recover the output using the inverse Fourier transform:

syms X H Y y wX = 2*sin(w/2)./w;H=(1/RC)./(j*w+1/RC);Y=X.*H;Y=ifourier(Y);ezplot(y,[-1 5]);axis([-1 5 0 1.5])

1/RC = 1

1/RC = 10

1/RC = 1

1/RC = 10

ECE 3163: Lecture 18, Slide 10

Ideal Filters• The process of rejecting particular frequencies or a range of frequencies is

called filtering. A system that has this characteristic is called a filter.• An ideal filter is a filter whose frequency response goes exactly to zero for

some frequencies and whose magnitude response is exactly one for other ranges of frequencies.

• To avoid phase distortion in the filtering process, an ideal filter should have a linear phase characteristic. Why?

• We will see this “ideal” response has some important implications for the impulse response of the filter.

• Lowpass

• Bandpass

• Highpass

• Bandstop

passbandfiltertheinωallfor)( dj teH

ECE 3163: Lecture 18, Slide 11

Ideal Linear Phase Lowpass Filter

BB

BBeeH

dtjj

,0

,)(

• PhaseResponse

• Consider the ideal lowpass filterwith frequency response:

• Using the Fourier transform pairfor a rectangular pulse, and applyingthe time-shift property:

• Is this filter causal?

• The frequency response of an idealbandpass filter can be similarly defined:

• Will this filter be physically realizable?Why?

dttB

cB

th (sin)(

• ImpulseResponse

elsewhere,0

,)( 21 BBe

eHdtj

j

ECE 3163: Lecture 18, Slide 12

Summary• Showed that the response of a linear LTI system to a sinusoid is a sinusoid at

the same frequency with a different amplitude and phase.

• Demonstrated how to compute the change in amplitude and phase using the system’s Fourier transform.

• Demonstrated this for a simple RC circuit.

• Generalized this to periodic and nonperiodic signals.

• Worked examples involving a periodic pulse train and a single pulse.

• Introduced the concept of an ideal filter and discussed several types of ideal filters.

• Noted that the ideal filter is a noncausal system and is not physically realizable. However, there are many ways to approximate ideal filters, and that is a topic known as filter design.