CHAPTER 3
ANALYSIS AND TRANSMISSION
OF SIGNALS
1
Aperiodic Signal:
Fourier Integral
Figure 3.1 Construction of a periodic signal by periodic extension of g(t).
Fundamental of Communication Systems
ELCT332
Fall2011
2
Figure 3.2 Change in the Fourier spectrum when the period T0 in Fig. 3.1 is doubled.
Fundamental of Communication Systems
ELCT332
Fall2011
3
The Fourier series becomes the Fourier integral in the limit as T0 →∞.
Fundamental of Communication Systems
ELCT332
Fall2011
4
Fourier integral
G(f): direct Fourier transform of g(t)
g(t): inverse Fourier transform of G(f)
Find the Fourier transform of
(a) e−atu(t) and (b) its Fourier spectra.
Dirichlet Condition
Linearity of the Fourier Transform (Superposition Theorem)
Fundamental of Communication Systems
ELCT332
Fall2011
5
Physical Appreciation of the Fourier Transform
•Fourier representation is a way of a signal in terms of everlasting sinusoids, or exponentials.
•The Fourier Spectrum of a signal indicates the relative amplitudes and phases of the
sinusoids that are required to synthesize the signal.
Analogy for Fourier transform.
Fundamental of Communication Systems
ELCT332
Fall2011
6
G(f): Spectrum of g(t)
Time-limited pulse.
Fundamental of Communication Systems
ELCT332
Fall2011
7
Transforms of some useful functions
Unit Rectangular Function
Rectangular pulse.
Fundamental of Communication Systems
ELCT332
Fall2011
8
Unit Triangular Function
Triangular pulse.
Fundamental of Communication Systems
ELCT332
Fall2011
9
Sinc Function
Sinc pulse.
Fundamental of Communication Systems
ELCT332
Fall2011
10
Example
(a) Rectangular pulse and (b) its Fourier spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
11
Example II
(a) Unit impulse and (b) its Fourier spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
12
Example III
(a) Constant (dc) signal and (b) its Fourier spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
13
Find the inverse Fourier transform of
(a) Cosine signal and (b) its Fourier spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
14
Sign function.
Fundamental of Communication Systems
ELCT332
Fall2011
15
Time-Frequency Duality
Dual Property
Near symmetry between direct and inverse Fourier transforms.
Fundamental of Communication Systems
ELCT332
Fall2011
16
Dual Property
Duality property of the Fourier transform.
Fundamental of Communication Systems
ELCT332
Fall2011
17
Time-Scaling Property
Time compression of a signal results in spectral expansion, and time expansion of the
signal results in its spectral compression.
The scaling property of the Fourier transform.
Fundamental of Communication Systems
ELCT332
Fall2011
18
Example
Prove that
and if
to find the Fourier transforms of
and
(a) e−a|t| and (b) its Fourier spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
19
Time-Shifting Property
Delaying a signal by t0 seconds does not change its amplitude spectrum. The phase
spectrum is changed by -2πft0 .
To achieve the same time delay, higher frequency sinusoids
must undergo proportionately larger phase shifts.
Question: Prove that
Physical explanation of the time-shifting property.
Fundamental of Communication Systems
ELCT332
Fall2011
20
Example
Find the Fourier transform of
Linear phase spectrum
Effect of time shifting on the Fourier spectrum of a signal.
Fundamental of Communication Systems
ELCT332
Fall2011
21
Frequency-Shifting Property
Multiplication of a signal by a
factor
shifts the
spectrum of that signal by f=f0
Amplitude Modulation
Carrier, Modulating signal,
Modulated signal
Amplitude modulation of a signal causes spectral shifting.
Fundamental of Communication Systems
ELCT332
Fall2011
22
Example:
Find the Fourier transform of the modulated signal
g(t)cos2πf0t in which g(t) is a rectangular pulse
Frequency division
multiplexing (FDM)
Example of spectral shifting by amplitude modulation.
Fundamental of Communication Systems
ELCT332
Fall2011
23
Bandpass Signals
(a) Bandpass signal and (b) its spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
24
Example:
Find the Fourier transform of a general periodic signal g(t) of period T0
(a) Impulse train and (b) its spectrum.
Fundamental of Communication Systems
ELCT332
Fall2011
25
Time Differentiation
Find the Fourier transform of the triangular pulse
Time Integration
Using the time differentiation property to find the Fourier transform of a piecewise-linear signal.
Fundamental of Communication Systems
ELCT332
Fall2011
26
Properties of Fourier Transform Operations
Operation
g(t)
G(f)
Superposition
g1(t)+g2(t)
G1(f)+G2(f)
Scalar multiplication
kg(t)
kG(f)
Duality
G(t)
g(-f)
Time scaling
g(at)
Time shifting
g(t-t0)
Frequency Shift
e j2f 0 t g(t)
g1(t)*g2(t)
Time convolution
G(f/a)/ | a |
e-j2ft0G(f)
G(f-f0)
G1(f)G2(f)
Frequency convolution g1(t)g2(t)
Time differentiation
Time integration
Fundamental of Communication Systems
G1(f)*G2(f)
( j 2f )n G( f )
(dn g)/dt n

t

G( f ) 1
 G(0) ( f )
j 2f 2
g ( x)dx
ELCT332
Fall2011
27
Signal Transmission Through a Linear System
H(f): Transfer function/frequency response
Signal transmission through a linear time-invariant system.
Fundamental of Communication Systems
ELCT332
Fall2011
28
Distortionless transmission:
a signal to pass without distortion
delayed ouput retains the waveform
Linear time invariant system frequency response for distortionless transmission.
Fundamental of Communication Systems
ELCT332
Fall2011
29
Determine the transfer function H(f),
and td(f).
What is the requirement on the bandwidth of
g(t) if amplitude variation within 2% and time
delay variation within 5% are tolerable?
(a) Simple RC filter. (b) Its frequency response and time delay.
Fundamental of Communication Systems
ELCT332
Fall2011
30
Ideal filters: allow distortionless transmission of a certain band of frequencies
and suppress all the remaining frequencies.
(a) Ideal low-pass filter frequency response and (b) its impulse response.
Fundamental of Communication Systems
ELCT332
Fall2011
31
Ideal high-pass and bandpass filter frequency responses.
Paley-Wiener criterion
Fundamental of Communication Systems
ELCT332
Fall2011
32
For a physically realizable system h(t) must be causal
h(t)=0
for t<0
Approximate realization of an ideal low-pass filter by truncating its impulse response.
Fundamental of Communication Systems
ELCT332
Fall2011
33
Butterworth filter characteristics.
The half-power bandwidth
•The bandwidth over which
the amplitude response
remains constant within 3dB.
•cut-off frequency
Fundamental of Communication Systems
ELCT332
Fall2011
34
Digital Filters
Sampling, quantizing, and coding
Basic diagram of a digital filter in practical applications.
Fundamental of Communication Systems
ELCT332
Fall2011
35
Linear Distortion
Magnitude distortion
Phase Distortion: Spreading/dispersion
Pulse is dispersed when it passes through a system that is not distortionless.
Fundamental of Communication Systems
ELCT332
Fall2011
36
Distortion Caused by
Channel Nonlinearities
Signal distortion caused by nonlinear operation: (a) desired (input) signal spectrum;
(b) spectrum of the unwanted signal (distortion) in the received signal; (c) spectrum of the received si
(d) spectrum of the received signal after low-pass filtering.
Fundamental of Communication Systems
ELCT332
Fall2011
37
Multipath Effects
Multipath transmission.
Fundamental of Communication Systems
ELCT332
Fall2011
38
Signal Energy: Parseval’s Theorem
Energy Spectral Density
Interpretation of the energy spectral density of a signal.
Fundamental of Communication Systems
ELCT332
Fall2011
39
Essential Bandwidth: the energy content of the components of frequeicies
greater than B Hz is negligible.
Figure 3.39 Estimating the essential bandwidth of a signal.
Fundamental of Communication Systems
ELCT332
Fall2011
40
Find the essential bandwidth where it
contains at least 90% of the pulse energy.
Fundamental of Communication Systems
ELCT332
Fall2011
41
Energy of Modulated Signals
Energy spectral densities of (a) modulating and (b) modulated signals.
Fundamental of Communication Systems
ELCT332
Fall2011
42
Autocorrelation Function
Determine the ESD of
Figure 3.42 Computation of the time autocorrelation function.
Fundamental of Communication Systems
ELCT332
Fall2011
43
Signal Power
Power Spectral Density
Limiting process in derivation of PSD.
Time Autocorrelation Function of Power
Signals
PSD of Modulated Signals
Fundamental of Communication Systems
ELCT332
Fall2011
44