Continuous-Time Signal Analysis: The Fourier Transform

Chapter 7Mohamed 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 FSFourier series (FS) allows us to represent periodic signal in term of sinusoidal or exponentials ejnωot.

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

xTo(t) ( ) ∑∞

−∞=

=n

tjnnT eDtx 0

0

ω

tjnT

TTn etx

TD 0

0

0

0

2/

2/0

)(1 ω−

−∫=

Link between FT and FS

( ) ( )txtx TT 00

lim∞→

=000 →⇒⇒∞→ ωT

xTo(t) x(t)

nD )(ωX

As T0 gets larger and larger the fundamental frequency ω0 gets smaller and smaller so the spectrum becomes continuous.

ω0 ω

)(10

0

ωnXT

Dn =

The Fourier Transform Spectrum

The Inverse Fourier transform:

∫∞

∞−

= ωωπ

ω deXtx tj)(21)(

The Fourier transform:

)()()(

)()(

ω

ω

ωω

ω

X

tj

eXX

dtetxX

∠

−∞

∞−

=

= ∫

The Amplitude (Magnitude) Spectrum The Phase SpectrumThe 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

ωωω

ω

ωω ω

−

∞−−

−=∠+

=

>+

== ∫

How does X(ω) relates to X(s)?

-aRe(s) if 1)(

1)(0

)(

0

>+

=

+−==

∞+−

∞−−∫

sasX

esa

dteesX tasstat

S-planes = σ + jω

Re(s)σ

jω

ROC

-a

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

Useful FunctionsUnit Gate Function

<=>

=

2/|| 12/|| 5.02/|| 0

τττ

τxxx

xrect

Unit Triangle Function

<−≥

=

∆

2/|| )/2(12/|| 0

τττ

τ xxxx

τ/2-τ/2

τ/2-τ/2

1

1

x

x

Useful FunctionsInterpolation Function

0for 1)(sincfor 0)(sinc

sin)(sinc

==±==

=

xxkxx

xxx

π

sinc(x)

x

Example

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

Answer

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

2/

ωτττωτ

τ

ω∫−

− == dtetrectX tj

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

What is the bandwidth of the above pulse?

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

1)()( ∫∞

∞−

− == dtetX tjωδω 1)( ↔tδ

Find the inverse FT of δ(ω).Answer

πωωδ

πω

21)(

21)( ∫

∞

∞−

== detx tj )(21 ωπδ↔

ExamplesFind the inverse FT of δ (ω - ω0).Answer

)(2 and )(2impulse shifted a isexponent complex a of spectrum theso

21)(

21)(

00

0

00

0

ωωπδωωπδ

πωωωδ

π

ωω

ωω

+↔−↔

=−=

−

∞

∞−∫

tjtj

tjtj

ee

edetx

Find the FT of the everlasting sinusoid cos(ω0t).Answer

( )

( ) [ ])()(21

21cos

00

0

00

00

ωωδωωδπ

ω

ωω

ωω

−++↔+

+=

−

−

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

ωωδπω

πωω

Examples

Find the FT of the unit impulse train

Answer

)(0

tTδ

∑

∑∞=

−∞=

∞=

−∞=

−=

=

n

n

n

n

tjnT

nT

X

eT

t

)(2)(

1)(

00

0

0

0

ωωδπω

δ ω

Properties of the Fourier Transform• Linearity:

Let and

then

( ) ( )ωXtx ⇔ ( ) ( )ωYty ⇔

( ) ( ) ( ) ( )ωβωαβα YXtytx +⇔+

• Time Scaling:

Let

then

( ) ( )ωXtx ⇔

( )

⇔

aX

aatx ω1

When a > 1 that leads to compression in the time domain which results in expansion in the frequency domain

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

Properties of the Fourier Transform• Time Reversal:

Let

then ( ) ( )x t X ω− ↔ −( ) ( )ωXtx ⇔

Example: Find the FT of e-a|t|

• Left or Right Shift in Time:

Let

then

( ) ( )ωXtx ⇔

( ) ( ) 00

tjeXttx ωω −⇔−

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

Property):

Let

then 00( ) ( )j tx t e Xω ω ω↔ −

( ) ( )ωXtx ⇔

• Multiplication by a Sinusoid (Amplitude Modulation):

Let

then

( ) ( )ωXtx ⇔

( ) ( ) ( ) ( )[ ]000 21cos ωωωωω −++⇔ XXttx

cosω0t is the carrier, x(t) is the modulating signal (message),x(t) cosω0t is the modulated signal.

Example: Amplitude Modulation

Example: Find the FT for the signal

-2 2

A

x(t)

ttrecttx 10cos)4/()( =

Amplitude Modulation

ttmt cAM ωϕ cos)()( =Modulation

]2cos1)[(5.0 cos)( ttmtt ccAM ωωϕ +=

Demodulation

Then lowpass filtering

Amplitude Modulation: Envelope Detector

Applic. of Modulation: Frequency-Division Multiplexing

1- Transmission of different signals over different bands2- Require smaller antenna

Transmitter Receiver

Properties of the Fourier Transform

• Differentiation in the Time Domain:

Let

then ( ) ( ) ( )n

nn

d x t j Xdt

ω ω↔

( ) ( )ωXtx ⇔

• Differentiation in the Frequency Domain:

• Let

then ( ) ( ) ( )n

n nn

dt x t j Xd

ωω

↔

( ) ( )ωXtx ⇔

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

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

Let

Then1( ) ( ) (0) ( )

t

x d X Xj

τ τ ω π δ ωω−∞

↔ +∫

( ) ( )ωXtx ⇔

• Convolution and Multiplication in the Time Domain:

Let

Then ( ) ( ) ( ) ( )x t y t X Yω ω∗ ↔

( ) ( )( ) ( )ω

ωYtyXtx

⇔⇔

)()(21)()( 2121 ωωπ

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• Duality ( Similarity) :

• Let

then ( ) 2 ( )X t xπ ω↔ −

( ) ( )ωXtx ⇔

x(t) X(ω)

Energy of a Signal• Parseval’s Theorem: since x(t) is non-periodic

and has FT X(ω), then it is an energy signals:

( ) ( )∫∫∞

∞−

∞

∞−

== ωωπ

dXdttxE 22

21

Real signal has even spectrum X(ω)= X(-ω), ( )∫∞

=0

21 ωωπ

dXE

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

Answer: ω = 12.7a rad/sec

Data Truncation: Window Functions

1- Truncate x(t) to reduce numerical computation 2- Truncate h(t) to make the system response finite and causal3- Truncate X(ω) 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?

)(*)(21)( and )()()( ωωπ

ω WXXtwtxtx ww ==

Truncated Signal

Truncation WindowOriginal Signal

Implications of Data Truncation

1- Spectral spreading2- Spectral leakage3- Poor frequency resolution

What happened if x(t) has two spectral components of frequencies differing by less than 4π/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 rate3- Small sidelobe peak

Data Truncation: Window Functions

Using Windows in Filter Design