Continuous-Time Signal Analysis: The Fourier Transform

Chapter 7

Mohamed Bingabr

Chapter Outline

• Aperiodic Signal Representation by Fourier Integral• Fourier Transform of Useful Functions• Properties of Fourier Transform• Signal Transmission Through LTIC Systems• Ideal and Practical Filters• Signal Energy• Applications to Communications• Data Truncation: Window Functions

Link between FT and FS

Fourier series (FS) allows us to represent periodic signal in term of sinusoidal or exponentials e jnot.

Fourier transform (FT) allows us to represent aperiodic (not periodic) signal in term of exponentials ejt.

xTo(t)

n

tjnnT eDtx 0

0

tjnT

T

Tn etxT

D 0

0

0

0

2/

2/0

)(1

Link between FT and FS

txtx TT 0

0

lim

000 T

xTo(t)xT(t)

nD )(X

As T0 gets larger and larger the fundamental frequency 0 gets smaller and smaller so the spectrum becomes continuous.

0

)(1

00

nXT

Dn

The Fourier Transform Spectrum

The Inverse Fourier transform:

deXtx tj)(2

1)(

The Fourier transform:

)()()(

)()(

X

tj

eXX

dtetxX

The Amplitude (Magnitude) Spectrum The Phase Spectrum

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

ExampleFind the Fourier transform of x(t) = e-atu(t), the magnitude, and the spectrumSolution:

)/(tan)(1

)(

0a if 1

)(

1

22

0

aXa

X

jadteeX tjat

How does X() relates to X(s)?

-aRe(s) if 1

)(

1

)(0

)(

0

sasX

esa

dteesX tasstat

S-planes = + j

Re(s)

j

ROC

-a

Since the j-axis is in the region of convergence then FT exist.

Useful FunctionsUnit Gate Function

2/|| 1

2/|| 5.0

2/|| 0

x

x

xx

rect

Unit Triangle Function

2/|| /21

2/|| 0

xx

xx

/2-/2

/2-/2

1

1

x

x

Useful FunctionsInterpolation Function

0for x 1)(sinc

for x 0)(sinc

sin)(sinc

x

kxx

xx

sinc(x)

x

ExampleFind the FT, the magnitude, and the phase spectrum of x(t) = rect(t/).

Answer

)2/sinc()/()(2/

2/

dtetrectX tj

The spectrum of a pulse extend from 0 to . However, much of the spectrum is concentrated within the first lobe (=0 to 2/)

What is the bandwidth of the above pulse?

ExamplesFind the FT of the unit impulse (t).Answer

1)()(

dtetX tj

Find the inverse FT of ().Answer

)(21

impulsean isconstant a of spectrum theso

2

1)(

2

1)(

detx tj

ExamplesFind the inverse FT of (- 0).Answer

)(2 and )(2

impulse shifted a isexponent complex a of spectrum theso

2

1)(

2

1)(

00

0

00

0

tjtj

tjtj

ee

edetx

Find the FT of the everlasting sinusoid cos(0t).Answer

)()(2

12

1cos

00

0

00

00

tjtj

tjtj

ee

eet

ExamplesFind the FT of a periodic signal.Answer

n

nn

tjnn

nn

nDX

TeDtx

)(2)(

FT ofproperty linearity use and sideboth of FT theTake

/2)(

0

000

ExamplesFind the FT of the unit impulse train Answer

)(0tT

n

n

n

n

tjnT

nT

X

eT

t

)(2

)(

1)(

00

0

0

0

Properties of the Fourier Transform• Linearity:Linearity:

• Let and Let and

thenthen

Xtx Yty

YXtytx

• Time Scaling:Time Scaling:

• LetLet

thenthen

Xtx

a

Xa

atx1

Compression in the time domain results in expansion in the frequency domain

Internet channel A can transmit 100k pulse/sec and channel B can transmit 200k pulse/sec. Which channel does require higher bandwidth?

Properties of the Fourier Transform• Time Reversal:Time Reversal:

• LetLet

thenthen ( ) ( )x t X Xtx

Example: Find the FT of eatu(-t) and e-a|t|

• Left or Right Shift in Time:Left or Right Shift in Time:

• LetLet

thenthen

Xtx

00

tjeXttx Example: if x(t) = sin(t) then what is the FT of x(t-t0)?

Time shift effects the phase and not the magnitude.

Example: Find the FT of and draw its magnitude and spectrum

|| 0ttae

Properties of the Fourier Transform• Multiplication by a Complex Exponential (Freq. Shift Multiplication by a Complex Exponential (Freq. Shift

Property):Property):

• LetLet

then then 00( ) ( )j tx t e X

Xtx

• Multiplication by a Sinusoid (Amplitude Modulation):Multiplication by a Sinusoid (Amplitude Modulation):

LetLet

thenthen

Xtx

000 2

1cos XXttx

cos0t is the carrier, x(t) is the modulating signal (message),x(t) cos0t is the modulated signal.

Example: Amplitude Modulation

Example: Find the FT for the signal

-2 2

A

x(t)

ttrecttx 10cos)4/()(

HW10_Ch7: 7.1-1, 7.1-5, 7.1-6, 7.2-1, 7.2-2, 7.2-4, 7.3-2

Amplitude Modulation

ttmt cAM cos)()( Modulation

]2cos1)[(5.0 cos)( ttmtt ccAM Demodulation

Then lowpass filtering

Amplitude Modulation: Envelope Detector

Applic. of Modulation: Frequency-Division Multiplexing

1- Transmission of different signals over different bands

2- Require smaller antenna

Properties of the Fourier Transform

• Differentiation in the Time Domain:Differentiation in the Time Domain:

LetLet

thenthen ( ) ( ) ( )n

nn

dx t j X

dt

Xtx

• Differentiation in the Frequency Domain:Differentiation in the Frequency Domain:

• LetLet

thenthen ( ) ( ) ( )n

n nn

dt x t j X

d

Xtx

Example: Use the time-differentiation property to find the Fourier Transform of the triangle pulse x(t) = (t/)

Properties of the Fourier Transform• Integration in the Time Domain:Integration in the Time Domain:

LetLet

ThenThen1

( ) ( ) (0) ( )t

x d X Xj

Xtx

• Convolution and Multiplication in the Time Domain:Convolution and Multiplication in the Time Domain:

LetLet

ThenThen ( ) ( ) ( ) ( )x t y t X Y

Yty

Xtx

)()(2

1)()( 2121

XXtxtx Frequency convolution

ExampleFind the system response to the input x(t) = e-at u(t) if the system impulse response is h(t) = e-bt u(t).

Properties of the Fourier Transform• Parseval’s TheoremParseval’s Theorem: : sincesince xx((tt)) is non-periodicis non-periodic

and has FTand has FT XX(()),, then it is an energy signals:then it is an energy signals:

dXdttxE22

2

1

Real signal has even spectrum XX(())= = XX(-(-)),,

0

21

dXE

ExampleFind the energy of signal x(t) = e-at u(t). Determine the frequency so that the energy contributed by the spectrum components of all frequencies below is 95% of the signal energy EX.

Answer: =12.7a rad/sec

Properties of the Fourier Transform• Duality ( Similarity) :Duality ( Similarity) :

• LetLet

thenthen ( ) 2 ( )X t x

Xtx

HW11_Ch7: 7.3-3(a,b), 7.3-6, 7.3-11, 7.4-1, 7.4-2, 7.4-3, 7.6-1, 7.6-6

Data Truncation: Window Functions1- Truncate x(t) to reduce numerical computation 2- Truncate h(t) to make the system response finite and causal3- Truncate X() to prevent aliasing in sampling the signal x(t)4- Truncate Dn to synthesis the signal x(t) from few harmonics.

What are the implications of data truncation?

)(*)(2

1)( and )()()(

WXXtwtxtx ww

Implications of Data Truncation1- Spectral spreading2- Poor frequency resolution3- Spectral leakage

What happened if x(t) has two spectral components of frequencies differing by less than 4/T rad/s (2/T Hz)?

The ideal window for truncation is the one that has 1- Smaller mainlobe width 2- Sidelobe with high rolloff rate

Data Truncation: Window Functions

Using Windows in Filter Design

=

Using Windows in Filter Design

=

Sampling TheoremA real signal whose spectrum is bandlimited to B Hz [X()=0 for || >2B ] can be reconstructed exactly from its samples taken uniformly at a rate fs > 2B samples per second. When fs= 2B then fs is the Nyquist rate.

n

ns

n

n

tjn

n

n

nXT

X

eT

txnTxtx

nTttxnTxtx

s

)(1

)(

1)()()(

)()()()(

Reconstructing the Signal from the Samples

n

n

n

nTtBnTxtx

nTthnTxtx

nTtnTxthtx

nTxthtx

XHX

)(2(sinc)()(

)()()(

)()(*)()(

)(*)()(

)()()(

LPF

Example

Determine the Nyquist sampling rate for the signal x(t) = 3 + 2 cos(10) + sin(30).

Solution

The highest frequency is fmax = 30/2 = 15 HzThe Nyquist rate = 2 fmax = 2*15 = 30 sample/sec

AliasingIf a continuous time signal is sampled below the Nyquist rate then some of the high frequencies will appear as low frequencies and the original signal can not be recovered from the samples.

LPF With cutoff frequency

Fs/2

Frequency above Fs/2 will appear (aliased) as frequency below Fs/2

Quantization & Binary Representation

111110101100011010001000

111110101100011010001000

43210-1-2-3

43210-1-2-3

nL 2

L : number of levelsn : Number of bitsQuantization error = x/2

x

x(t)

1minmax

L

xxx

ExampleA 5 minutes segment of music sampled at 44000 samples per second. The amplitudes of the samples are quantized to 1024 levels. Determine the size of the segment in bits.

Solution

# of bits per sample = ln(1024) { remember L=2n }n = 10 bits per sample# of bits = 5 * 60 * 44000 * 10 = 13200000 = 13.2 Mbit

Problem 8.3-4Five telemetry signals, each of bandwidth 1 KHz, are quantized and binary coded, These signals are time-division multiplexed (signal bits interleaved). Choose the number of quantization levels so that the maximum error in the peak signal amplitudes is no greater than 0.2% of the peak signal amplitude. The signal must be sampled at least 20% above the Nyquist rate. Determine the data rate (bits per second) of the multiplexed signal.

Discrete-Time Processing of Continuous-Time Signals

Discrete Fourier Transform

dtetxX tj )()(

n

njenxT

X )(1

)(

1-N

0n

/2)()( NknjenxkX

k

Link between Continuous and Discrete

dtetxX tj )()(

1-N

0n

2

)()(n

N

kj

enxkX

x(t) x(n)Sampling Theorem

x(t) Laplace TransformX(s) x(n) X(z)

z Transform

x(t) X(j) x(n) X(k)Fourier Transform Discrete Fourier Transform

dtetxsX st)()(

n

n

nznxzX )()(

t

x(t)

Continuous Discrete

x(n)

n