Lecture Set 4

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ECE 219 : Signal, System & Control

Chapter 4: Continuous-Time Signal Analysis: The Fourier Transform

Navneet Upadhyay Dept. of Electronics & Communication Engineering

The LNM Institute of Information Technology, Jaipur

Link Between Fourier Transform (FT) and Fourier Series (FS) (I)

Limiting process: An a-periodic signal can be expressed as a continuous sum (integral) of everlasting exponentials (or sinusoids).

xTo(t) x (t)

txtx TT 0

0

lim

i.e. pulses in periodic signal repeats after an infinite interval.

FS representing xT0(t) will also represent x(t) in the limit T0→∞

where

and

The Exponential FS of xT0(t)

Link Between Fourier Transform (FT) and Fourier Series (FS) (II)

i.e. Dn are (1/T0) times the samples of X(ω), which are uniformly spaced at intervals of ω0

0000 nDT

Doubling T0, halves the fundamental frequency ω0 and envelop 1/T0 X(ω), but not shape

Integrating xT0(t) over (-T0/2, T0/2) is the same as integrating x(t) over (-∞.∞)

To see the nature of the spectrum as T0 ↑, define X(ω) as a

continuous function of ω

Relation between Dn and X(ω)

i.e. spectrum is so dense that the spectral component are spaced at zero intervals, and amp. of each component is zero.

Link Between FT and FS (III)

From Eq. (1) and Eq. (4)

000 T replace ω0 by more appropriate notation Δω, Δω=2π/T0

xT0(t) is the sum of everlasting exponentials of frequencies 0,+Δω,+2Δω,+3Δω,

(FS). Amount of component of frequency nΔω is . In the limit

and

,....3,2,,0

n

2

nX

00 T txtxT 0

tjeX Area under the function . Thus

dtetxtxFX tj

)()(

The signal is approximated by sum of complex exponentials with Dn = X(n∆ω). [EFS] FS becomes the Fourier integral in the

limit of T0→∞

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The Fourier Transform Spectrum

The Inverse Fourier transform:

deXXFtx tj)(2

1)( 1

The Fourier transform:

The Amplitude (Magnitude) Spectrum The Phase Spectrum

i.e. if x(t) is a real function of t, then X(ω) and X(-ω) are complex conjugate.

The amplitude spectrum is an even function and the phase is an odd function.

)()()(

sincos)()(

X

tj

eXX

tdttxjtdttxdtetxX

dtetxX tj

)()(

*

*)(

XX

XX

Transform pair ( ) ( )x t X

For all ω

Summary: Definition of Fourier Transform

FS is used to represent periodic signal in term of sinusoidal or exponentials ejn0t.

Fourier Transform is used to represent a-periodic (not periodic) signal in term of exponentials ejt. The forward and inverse FT are defined for a-periodic signal as:

Existence of the Fourier Transform (I) A signal x(t) is said to have a Fourier transform in the ordinary

sense if the above integral converges

Fact: The integral does converge if

1. the signal x(t) is “well-behaved”

2. and x(t) is absolutely integrable, namely,

well behaved means that the signal has a finite number of

discontinuities, maxima, and minima within any finite time interval

then X(ω) exists for every frequency ω and is continuous.

Fact: if x(t) has finite energy, i.e.,

then X(ω) exists for “most” frequencies ω and has finite energy.

Existence of the Fourier Transform (II) Fact: if x(t) is periodic and has a Fourier series, then

is a weighted sum of impulses in frequency domain.

Consider the signal

| ( ) |x t dt dt

Clearly x(t) does not satisfy the first requirement since

Therefore, the constant signal does not have a Fourier transform in the

ordinary sense

Consider the signal

1tx

tuetx at

t

t

tjatjatja

tjat

eja

dtedte

dtetuex

0

0

0 10

If a<0, FT, X(ω) does not exist

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Fourier Transform Examples

One-sided exponential decay is defined by e−atu(t) with a > 0:

The Fourier transform of one-sided decay is:

Since x(t) has finite area, its transform is continuous.

x(t) is real but its transform is complex valued.

Fourier integral of e−atu(t) does not converge for a < 0. Hence FT of e−atu(t) does not exixt for a < 0 (growing exponential).

If a=0, x(t)=u(t), and X(ω) DOES NOT EXIST IN ORDINARY SENCE.

Fourier Transform Examples (Cont’d) We can rationalize X(ω).

We can better picture X(ω) using polar representation

FT produce continuous frequency spectra while FS produce discrete line spectra

with nonzero values at specific frequencies.

Fourier Transform Examples (Cont’d)

Fourier transform at ω = 0:

is the area under x(t), called the DC value.

For x(t) = e−atu(t), X(0) =1/a is the only real value and largest in magnitude.

For one-sided decay, X(−ω) = X∗(ω), complex conjugate of X(ω).

This is true for all real-valued signals.

We need only positive frequencies for real-valued signals.

Define Three Useful Functions

A unit rectangular window (also called a unit gate) function rect(x):

A unit triangle function Δ(x):

Interpolation function Sinc(x): The sinc function is very important. Sadly, it has two definitions.

2/|| 1

2/|| 5.0

2/|| 0

x

x

xx

rect

2/|| /21

2/|| 0

xx

xx

x

xxcor

x

xxc

sinsin

sinsin

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More about Sinc(x) function

sinc(x) is an even function of x.

02sin nc ;If n ≠ 0

sinc(x) = 0 when sin(x) = 0 except when

x=0, i.e. x = ±π,±2π, ±3π…..

sinc(0) = 1 (derived with L’Hôpital’s rule)

1sin

0sin lim0

x

xc

x

;1sin 2xdxc Thus sinc(x) has a finite energy FT

dtcxsin

sinc (x) is a band-limited pulse with no frequency content for f >1/2.

Fourier Transform of x(t) = rect(t/τ) Evaluation:

Since rect(t/τ) = 1 for -τ/2 < t <τ/2 and 0 otherwise

Fact: every finite width pulse has a transform with unbounded frequencies.

dX

2

2

tf

Amplitude spectrum

Phase spectrum

Fourier Transform of unit impulse x(t) =δ(t)

Using the sampling property of the impulse, we get:

IMPORTANT – Unit impulse contains COMPONENT AT EVERY FREQUENCY.

dtett tj)()]([F 10

t

tje

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Inverse Fourier Transform of δ(ω)


Spectrum of a constant (i.e. d.c.) signal x(t)=1 is an impulse 2πδ(ω).

1Ft

de tj

2

1

22

tjtj ed

e

090,2

1 tt

Inverse Fourier Transform of δ(ω-ω0)


Spectrum of an everlasting exponential ejω0t is a single impulse at ω = ω0.

Fourier Transform of everlasting Sinusoid cosω0t

Remember Euler formula:

Spectrum of cosine signal has two impulses at positive and negative frequencies.

Use result from previouss slide:

The impulse pairs at + and − frequencies correspond to two

phases.

u(t) is not absolutely integrable.

Approach this by considering u(t) to be a decaying exponential e-atu(t) in the limit a→0

tuetu at

a

0lim

da

a22

220lim

a

a

a

j

U1

when a→0, function approaches to

1 for all ω≠0 and its area π

concentrate at a single point ω =0

FT of Unit Step Function u(t)

Thus

ja

a

aj

a

aU

jaUtuF

a

a

a

1lim

lim

1lim

220

22220

0

Area under this function is π regardless the value of a

()

0

|U()|

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sgn(t) in terms of u(t)

tutut sgn

Not absolutely integrable, approximate sgn(t) to be sum of exponentials e-atu(t)–eat u(t) in the limit a→0

This looks like

FT of sgn(t)

0,2

0,2

0tansgn,

2sgn

2sgn

limsgn

1

0

jt

tuetuet atat

a

Example: FT of u(t) interms of sgn(t)

)sgn(2

1

2

1ttu

Transform of the signum function (or sign function) is

Therefore

let a→0 the exponential function resembles more and more closely the signum function.

Fourier Transform of any periodic signal

Fourier series of a periodic signal x(t) with period T0 is given by:

Take Fourier transform of both sides, we get:

n

tjn

neDtxX 0)]([)( FF

n

tjn

n eD ][ 0F

n

n nD )(2 0

n

n nD )(2 0

n

n nDX )(2 0

0

0

2)( 0

TeDtx

tjn

n

n

The FT of a periodic function consists of a sequence of equidistant impulses located at the harmonic frequencies of the function.

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Fourier Transform of a unit impulse train

Consider an impulse train:

The Fourier series of this impulse train :

Therefore using results from the previous slide, we get:

Fourier Transform Table (1)



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Linearity & Conjugate Properties of FT

If and

If

If is real

then

then

then

Time-Frequency Duality (Symmetry) of Fourier Transform

Near symmetry between direct and inverse Fourier transforms:

If F is the system that produces the Fourier transform, then

The forward transform results in the reversal of the inverse transform. This is called the principle of duality.

xtXFXtxF 2)]([

Duality (Symmetry) Property of FT

Proof: From definition of inverse FT (previous slide), we get

If

then

Hence

Change t to ω yield, and use definition of forward FT, we get:

Duality Property Example

Consider the FT of a rectangular function:

By duality, Fourier transform of sinc x is Π(f).

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Scaling Property of FT

That is, compression of a signal in time results in spectral expansion (and reduction in amplitude), and vice versa

If

then for any real constant a,

The sharper the pulse the wider the spectrum

Time Shifting Property of FT

Consider a sinusoidal wave, time shifted:

If

then

Obvious that phase shift increases with frequency (t0 is constant).

Frequency-Shifting (Modulation)

Property of FT

If

Then

)(Xtx

)( 00

Xetxj

dteetxetx tjtjtj

00 )(])([F

dtetxtj

)( 0)(

)( 0 X

Proof:

Frequency-Shifting Example

Find and sketch the Fourier transform of the signal x(t) cos10t where x(t) = rect(t/4).

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Convolution Properties of FT

Let H(ω) be the Fourier transform of the unit impulse response h(t), i.e.

If

Then

and

Applying the time-convolution property to y(t) = x(t)*h(t), we get:

That is: the Fourier Transform of the system impulse response is the system Frequency Response

Proof of the Time Convolution Properties

By definition

The inner integral is Fourier transform of x2(t-τ), therefore we can use time-shift property and express this as X2(ω)e-jωτ .

Frequency Convolution Example

Find the spectrum of x(t) = cos10t where x(t) = rect(t/4). Using convolution property.

Time Differentiation Property of FT

If

Then

)(Xtx

)(Xjtxdt

d

Proof: dtetxtx

dt

dF tj

)('

dtetxjetx tjtj )()(

)(Xj Generalize case:

If )(Xtx

Then )( Xjtxnn

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Time Integration Property of FT

If

Then

and

Summary of Fourier Transform Operations (1)

Summary of Fourier Transform Operations (2)

ECE 219 : Signal, System & Control

Signal Transmission & Windowing Effects

Navneet Upadhyay Dept. of Electronics & Communication Engineering

The LNM Institute of Information Technology, Jaipur

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Signal Transmission Through LTIC Systems

H

thLTI System,

Input signal Output signal

thtxty *

XHY

tx

X

X

YH )(

= Impulse response of the system

=Frequency response of the system

H

th

Example

Find the zero-state response of a stable LTI system with transfer function

and the input is x(t) = e-t u(t).

The FT of input x(t) is:

Since the system is stable, therefore H(jω) = H(ω). Hence

Therefore

Using partial fractions, we get:

Signal Distortion during Transmission

In polar form

H

thLTI System,


thtxty *

XHY

tx

X

X

YH )(

HjXjYj eHeXeY )(

H

During transmission, input signal spectral component of frequency ω

is modified in amplitude by a factor and shifted in phase by

an angle

H

= Amplitude response of the system

= Phase response of the system

H

H

Distortion Types In applications such as signal amplification or message signal

transmission over a communication channel, we require that the output waveform be a replica of the input waveform

In such cases we need to minimize distortion

Two Types of Distortion:

1. Amplitude response is not constant over a frequency band (interval) of interest amplitude distortion

2. Phase response is not linear over a frequency band of interest phase distortion

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Distortionless Transmission

Transmission is said to be distortionless if input signal and output signal have identical wave shapes within a multiplicative constant and a Time delay constant .

H

thLTI System,


dttGxty

Y

tx

X

dtj

d

GeHX

Y

ttGxty

FR required for distortionless

transmission

Constant G is used

for change in amplitude

Constant td accounts

for delay in transmission

GH

i.e. Amplitude response must be constant and Phase response must be a linear function of ω with slope -td and intersect at zero.

i.e.

dtH

Same shape as input, Different

magnitude with delay

linear phase means that every spectral component is

delayed by td seconds.

Therefore, a distortionless transmission needs a flat amplitude response and a linear phase response (i.e. phase is not only a linear function of ω but also

pass through origin at ω=0):

dtH

slope -td

Phase linearity measure: The phase distortion of a

linear system can be characterized using group delay. plot the slope of as a function of ω H

Hd

dtg

If tg (ω) is constant, signal is delayed by tg (assuming constant H(ω)).

If tg (ω) is not constant constant, signal is distorted.

tg (ω) is known as Group delay or Envelope delay.

Human ears are sensitive to amplitude distortion, but not phase distortion.

Human eyes are sensitive to phase distortion, but not (so much) amplitude

distortion.

Bandpass Systems & Group Delay

If one applies an input z(t) = x(t) cosωct, then the output y(t) is:

That is, the output is the delayed version of input z(t) and the output carrier acquires an extra phase φ0.

The envelope of the signal is therefore distortionless.

Consider a bandpass system with amplitude and phase characteristics as shown:

H

Example

A signal z(t) shown below is given by z(t) = x(t)cosωct where ωc=2000π.

The pulse x(t) is a lowpass pulse of duration 0.1sec and has a bandwidth of

about 10Hz. This signal is passed through a filter whose frequency response

is shown below. Find and sketch the filter output y(t).

Spectrum Z(ω) is a narrow band of bandwidth of 20Hz centered around 1kHz (=f0).

The gain at 1kHz (=f0) is 2. The group delay is:

The vertical axis intercept of phase response is

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Parseval’s Theorem: Energy Preserving

The energy of a signal x(t) in time or frequency domain:

Proof:

Energy in time domain = Energy in Frequency domain

Parseval’s Theorem

Band Energy:

B

B

w

w

x dXE

2

2

1

Energy Spectral Density of a signal

The energy over a small frequency band Δω (Δω→0) is:

Total energy is area under the curve of vs ω (divided by 2π)

Energy Spectral Density of a REAL Signal

If x(t) is a real signal, then X(ω) and X(-ω) are conjugate

This implies that X(ω) is an even function. Therefore

Consequently, the energy contributed by a real signal by spectral components between ω1 and ω2 is (i.e. Band Energy):

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Example

Find the energy E of signal x(t) = e-at (t). Determine the frequency W (rad/s) so that the energy contributed by the spectral component from 0 to W is 95% of the total signal energy E.

Take FT of x(t):

By Parseval’s theorem:

Energy in band 0 to W is 95% of this, therefore:

Note: For this signal, 95% of energy is in small frequency band from 0 to 12.706a rad/s or (2.02a Hz). All remaining bands from 12.706a rad/s to ∞ contribute only 5% of energy.

Bandlimited Signals

A signal x(t) is said to be band-limited if its Fourier transform

X(ω) is zero for all |ω|>2πB, where B is some positive number, called the bandwidth of the signal.

It turns out that any band-limited signal must have an infinite duration in time, i.e., band-limited signals cannot be time limited

If a signal x(t) is not band-limited, it is said to have infinite bandwidth or an infinite spectrum.

Time-limited signals cannot be band-limited and thus all time-limited signals have infinite bandwidth

Lecture Set 4

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