Continuous-Time Signal Analysis: The Fourier Transformmbingabr/Signals_Systems/...Continuous-Time...
Transcript of Continuous-Time Signal Analysis: The Fourier Transformmbingabr/Signals_Systems/...Continuous-Time...
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