My Introduction
• Syed Rizwan-ul-Hasan
– Asst. Professor
– B.Sc (Industrial Tech), MUET
– BS (Electronics / Computer Sc), USA
– MS (Computer Engineering), SSUET
– (PhD) Computer Engineering ,SSUET
– Experience; 27 years, which include 14 years
of field experience in USA, Saudi Arab and
Pakistan ; 13 years of academic.
My Introduction …
 Research Interest
– Multi-Carrier Communication
• MCCDMA(Multi-Code Code Division Multiple Access)
• OFDM(Orthogonal Frequency Division Multiplexing)
– Peak Power control
 Office contact
– srhasan@ssuet.edu.pk
– Room no. BT-9
– Phone; 4988000-2 / ext 252
Communications Systems
CE-308
5TH. SEMESTER
COMPUTER ENGINEERING DEPARTMENT
COURSE OUTLINE
Introduction to Signals & Comm.
• Week 1
– Signal & Its Classification
– Important Signals
– Operations On Signal
– Power Signals
• Week 2
• Communication System Components
• Types of Communication Systems
• Application of Communication System
– SNR, Noise, Channel Bandwidth, Data rate
– Basic Filter Types
– Nyquist Theorem, Shannon’s Theorem
FOURIER ANALYSIS
• Week 3
–
–
–
–
–
Fourier series for Periodic Signals
Fourier Transform
Parseval’s Theorem
Application of Fourier Transform
Power Spectrum Density Function
• Week 4
• Properties of Fourier Transform
• Channel & their types with distortions
FOURIER ANALYSIS…
• Week 4 …
– Convolution Operation
– Impulse Response & T.F of Channels/Filters
AMPLITUDE MODULATION AM
• Week 5
– Double Side Band – Suppressed Carrier(DSB-SC)
– Modulators & Demodulators of DSB-SC,
– DSB Single Band Modulation
– Hilbert Transform
• Week 6
– Modulators & Demodulators of SSB
– Telephone Channels using SSB
– Vestigial Sideband VSB & Application
– Carrier Acquisition & Phase locked loop PLL
INTRODUCTION TO
DIGITAL COMMUNICATION (Week 7 & 8)
• Digital Signal, Digital Communication
Component
• Sampling & Quantization
• AWGN probability Density Function
BASEBAND DIGITAL COMMUNICATION
(Note; Week 9 is Midterm)
• Week 10
– Detection of Baseband Signals
– Detection 7 likelihood test
• Week 11
– Vector representation of Signals
– Vector Analysis of Signals & noise
• Week 12
– Error probability & Error performance curves
– Match filters & Correlation
BAND PASS SIGNALS
• Week 13
– ASK, PSK & FSK
– Detection of Binary Band Pass Signals
– Detection of Multiple Band Pass Signals
• Week 14
– Different PSK Implementation
– Non Coherent & Coherent Detection of Signals
– Orthogonality of Signals
• Week 15
– Error performance for different modulation Schemes
– M-ary Signaling & Performance
• Week 16 (Revision)
Text and Ref. Books
• Text Books
– Modern Digital and Analog Communication
systems, 3rd. Edition by B.P.Lathi
– Digital communication fundamental and
Application, 2nd. Edition by Bernard Skalar
• Reference Books
– Signals & Systems, 2nd. Edition by Alan V.
Oppenheim and Alan S. Willsky with S. Hamid
Nawab
Reference books continue
– Digital Signal Processing, Principles, Algorithms
and Application by John G. Proakis and Dimitris
G. Manolakis
Marks distribution
Total marks 100
• Theory (20)
– Attendance (2)
– Quizzes and Assignments (3)
– Midterm (15)
• Lab (20)
– Attendance (4)
– Lab manual (8)
– Final test (8)
• Semester Exam (60)
Signals and Systems
(Week 1)
• In the fields of communications, signal
processing, and in electrical engineering
more generally, a signal is any time-varying
quantity.
• In the physical world, any quantity
measurable through time can be taken as a
signal.
Signals and Systems ….
• Whereas the systems respond to particular
signals by producing other signals or some
desired behavior.
• For e.g. In Electrical circuit, voltages and
currents as a function of time are signals, and
a circuit is itself a system.
• Another example a robot arm is a system,
whose movements are the response to
control input signal.
Signals and Systems…
• The concept of signals and systems arise in a
wide variety of fields.
• Areas – communications, aeronautics and
astronautics, circuit design, acoustics,
seismology, biomedical engineering, energy
generation and distribution systems,
chemical process control, and speech
processing.
• Graphical rep. of a signal is shown in fig. 1.
Fig.1.Graphical rep. of Signal
Mathematical rep. of a signal
• S(t) = 8t ------------------------------------ (1)
• The above equation will given only linear relationship
between signal S(t)and time ‘t’.
• S(t) = A Sine(2∏ft + ø) ------------------(2)
• However, equation 2 is a sine wave or sinusoidal S(t)
with parameters such as amplitude ‘A’, frequency ‘f’,
time ‘t’ and phase angle ‘ø’.
• Hence equation 2 is useful in analysis of for e.g. speech
signal, because it gives complete information about the
signal compare to equation 1.
• Sine wave or Sinusoid
Math. Rep of a Signal …
A, the amplitude, is the peak
deviation of the function from
its center position.
ω or 2∏f , the angular
frequency, specifies how many
oscillations occur in a unit time
interval, in radians per second
ɸ, the phase, specifies where
in its cycle the oscillation
begins at t = 0.
The sine wave is important in
physics because it retains its
wave shape when added to
another sine wave of the same
frequency and arbitrary phase.
– Fig. 2, Phase diff of 900,
Cosine wave leads by 900
Math. Rep of a Signal…
• It is the only periodic waveform that has this
property. This property leads to its
importance in Fourier analysis and makes it
acoustically unique.
• Cosine wave said to be sinusoidal, because
cosx = sinx(x + ∏/2), which is also a sine wave
with a phase difference of ∏/2.
Fig. 3.Occurrence of a Sinusoid
Classification of signals
Periodic Signal
• A periodic signal repeat
itself on a fixed interval of
length 2∏ as shown in
Fig. 4 . Sinusoidal signals
are examples of it.
• S(t) = S(t + T)
Non-periodic Signal
• While non-periodic signal
does not repeat itself as
shown in Fig. 4.
Fig. 4 Classification of Signals…
Periodic Sig., with two Periods
(a) A periodic signal with period
T0
(b) An aperiodic signal
Figure 3
Non-Periodic Signal
Classification of signals…
Continuous time
• Continuous time signal or
analog signal are defined for
every value of time and
they take on values in the
continuous interval ( x, y)
Discrete time
• Discrete time signal are
defined only at certain
specific values of time.
These time instant needs
not be equidistant. For e.g.
n=0, +1, +2,…..
Classification of Signal…
Analog vs. Discrete
• Same as Continuous vs. Discrete
Classification of Signals
Deterministic
• Signals whose values at any
instant ‘t’ is known from
their analytical or graphical
description are called
deterministic signals, as
shown in Figure 5.
• This type of signal convey
no information.
• Example is electronic
circuits based on Ohm’s law.
Random
• Random means uncertainty. If the signal or
message has un-certainty, it
means, there is information
as shown in Figure 6.
• Noise signals that perturb
information are example of
random signals.
Classification of signals…
Deterministic Signal Fig.5 Random Signal Fig.6
Classification of Signals…
Causal signals are signals that
are zero for all negative time
Anti-causal are signals that
are zero for all positive time.
Classification of signal…
• Anti-causal signals have nonzero values in both positive and
negative time.
Classification of signals…
A signal fe(t) is said to be
even, if it is identical to its
time reversed counter part
fe(t)=fe(-t)
(a) An even signal
A signal fo(t)is said to be odd,
if it is 0 at t=0; fo(t)=-(fo(-t))
(b) An odd signal
Classification of signals…
• Continuous time complex exponential (Fig.7)
– If σ is +, then as t increase s(t) is growing
exponential.
– Examples; Chain reaction in atomic explosion,
complex chemical reaction etc…
– If σ -, then s(t) is a decaying exponential.
– Examples; radioactive decay, Responses of RC
circuit etc…
Classification of signals…
Fig. 7 Exponential signals
(a) If σ is negative,
we have the case of a
decaying exponential
window
(b) If σ is positive, we (c) If σ is zero, we
have the case of a
have the case of a
growing exponential constant window.
window.
Signal Operations
(Time Shifting of a Signal)
Note; Subtracting from the time variable ‘t’ will cause delay (move the signal to the right)
, while adding advance (to the left).
Signal Operations…
Time Scaling
Time scaling compresses or dilates a signal by
multiplying the time variable by some quantity.
If that quantity is greater than one, the signal
becomes narrower and the operation is called
compression,
while if the quantity is less than one, the signal
becomes wider and is called dilation.
Signal operations…
Time Reversal
A natural question arises about time scaling is:
What happens when the time variable is multiplied
by a negative number? The answer to this is time
reversal. This operation is the reversal of the time
axis, or flipping the signal over the y-axis.
Signal Energy and Power
• In many applications the signals we consider are directly
related a physical quantities capturing power and energy in
physical system.
p(t )  v(t )i (t )
1 2
p (t )  v (t )
R
Signal Energy and Power …
• The total energy expended over the time
interval t1  t  t2 is
t2
t2
•
1 2
•
 p(t )dt   v (t )dt
t1
t1
R
• Similarly, for average power
t1
1
1 2
p(t ) 
v (t )dt

(t1  t2 ) t R
2
Signal Energy
• The previous equations are basic, but they may
be applied to continuous signal for evaluating
Energy and Power, with the assumption that
R=1.
• We may consider the area under the signal S(t)
as a possible measure of its size, because it takes
amplitude and duration both.
• This could be a improper measure, due to its
large size and its positive and negative value,
which cancel each other
Signal Energy…
• This indicates a signal of small size.
• This difficulty can be corrected by defining signal size as
S2(t), which is always positive.
• This measure is called Signal Energy. For a real valued and
complex signal:-
•

S e   s (t ) dt
2


S e   | s (t ) | dt
2

Signal Energy…
• The Signal Energy must be finite for it to be
meaningful measure of the signal size.
• A necessary condition for the energy to be finite is
that the signal amplitude →0 as |t|→∞, otherwise
the integral will not converge.
Signal Power
• If the amplitude of a signal does not →0 as |t|→∞,
the signal energy is infinite. In such a case a better
measure of a signal size would be a, average power
Ps , defined for a real and complex signal:-
1 T /2 2
Ps  lim  s (t )dt
T  T T / 2
2
lim 1
Ps 
| s(t ) | dt

T   T T / 2
T /2
Communications Systems
(Week 2)
• Communication means to share information;
To transmit or receive information or data.
• Communication may be established between
people, Computer to Computer, near distance
far distance (Telecommunication)
• The communication system , consist of
different components which helps in
transmitting or receiving data, between two
entities or devices.
Communications Systems…
• As shown in Fig. 8; Communication system
components consists of Source, Input
transducer , Transmitter, Channel, Receiver,
and Output transducer.
• Source originates the message, for e.g. voice,
picture or data. If the data is non-Electrical
such as voice, then it must be converted to
Electrical signal by Transducer known as
baseband signal or message signal.
Fig. 8 Communication System
Input
Message
Transmitted
Signal
Input
Transducer
Transmi
tter
Received
Signal
Channel
Receiver
Input Signal
Noise
Output
Signal
Output
Message
Output
transdu
cer
Communications Systems…
• The transmitter modifies the baseband signal
for efficient transmission.
• A transmitter consists of one or more subsystems, a sampler, a quantizer, a coder and a
modulator.
• A channel is a medium, such as coaxial cable,
a waveguide, an optical fiber, or a radio linkthrough which the transmitter output is sent.
Communications Systems…
• The receiver reprocesses the signal from the
channel by undoing the signal modification
made at the transmitter and the channel
• Finally, the receiver output is fed to the output
transducer, which convert back the electrical
signal to its original form i.e. the message
signal
• The Signal is distorted by Channel and Noise,
which are random and unpredictable
Communications Systems…
• That comes from external and internal sources
• External sources comes from nearby channel,
lightning, tube light electrical equipment etc…
• Internal noise results from thermal motions of
electrons in conductors.
• The signal to noise ratio is defined as the ratio
of the signal power to the noise power. The
channel distort the signal and the noise
accumulates along the path.
Communications Systems…
• The signal strength decreases while the noise
level increases with distance from the
transmitter.
• Thus SNR is continuously decreasing along the
channel and amplification of the noisy signal
make no use.
• For good results SNR, supposed to be high. In
other words signal value, should be high
compare to noise.
Digital Communication System
• Block diagram of CDMA system is shown in next
slide. Since the human speech is the analog signal,
so it has to be first converted into digital form. This
function is performed by the source encoding
module.
• After the source information is coded into a digital
form, redundancy needs to be added to this digital
message or data. This done for error control and
power reduction.
Digital Communications Systems
Reconstructed
Speech
Speech
• Block diagram
Source
Encode
Channel
Encode
Multiple
Access
Modulate
Transmitter
Source
Decode
Channel
Decode
Multiple
Access
DeModulate
Receiver
Digital Communication system…
• Thereafter, signal is further transformed to allow
access to multiple users. Multiple access by
different users means to the sharing of a common
resource i.e. RF (Radio Frequency) spectrum.
• The purpose of the modulator is to shift the
message or data to the carrier frequency (high
frequency), because message or data does not have
enough strength to go far distance.
• At the receiver end the purpose of the demodulator
is to recover original signal. In other words at the
receiver end the reverse operation is performed.
Application of Comm. System
• Telephone Exchange
• Wireless Communication
Digital vs. Analog
Digital
• More immune to channel
noise and distortion
• Regenerative repeaters
for noise free signal.
• Digital hardware
implementation is
flexible in reconfiguring
the hardware simply by
changing the program.
• Accuracy
Analog
• Less immune to noise
and distortion
• Not possible in analog
communication
• Fixed, and need to
redesign for new
hardware
• Difficult to control the
accuracy
Digital vs. Analog
Digital
• Coding to yield low error
rate
• Efficient in exchange of SNR
for Bandwidth
Analog
• Less exchange of SNR for
Bandwidth
SNR, Bandwidth, Data Rate
• The fundamental parameters that control the rate
and quality of information transmission are the
channel bandwidth and the signal power S.
• The bandwidth (BW) of a channel is the range of
frequencies that it can transmit with reasonable
fidelity .
OR
• Difference between the highest and the lowest
frequencies in the specific range of frequencies.
• Example; Voice frequency range of 300hz. To
3300hz. Thus the voice bandwidth (BW) or Pass
band is 3000hz wide
SNR, Bandwidth, Data Rate…
• Role of BW, If we want to increase the speed of
information transmission by time compression of
the signal lets say by a factor of 2.
• The signal can be transmitted in half time.
• Frequencies and channel BW must also be doubled.
• Thus the rate of information transmission is directly
proportional to channel BW.
• The signal power S plays a dual role in information
transmission. Increase S reduced the effect of
channel noise and we received accurate data.
SNR, Bandwidth, Data Rate…
• Signal to Noise Ratio (SNR) means, the higher
(strength) the value of the signal, compare to Noise,
the quality of the signal would be better over a
longer distance.
• However, a certain minimum SNR is necessary for
communication.
• The second role of signal power ‘S’ is not as
obvious, but it is important. The BW and ‘S’ are
exchangeable.
SNR, Bandwidth, Data Rate…
• For example if we increase the BW (by adding
redundant bits to the message for reliability) the
signal power S would reduced and vice versa.
• Another example; telephone channel has limited
BW, but requires lot of power.
• So, we have to trade in between BW & S.
• Since SNR is proportional to S, Therefore SNR and
BW are exchangeable.
• In practice, increasing BW to reduce signal power S
is followed and is rarely vice versa.
Shannon’s and Nyquist Theorem
• The limitation imposed on communication by
the channel bandwidth BW and the SNR is
highlighted by Shannon’s equation:• C  B log (1  SNR)
2
• Where ‘C’ is the channel capacity in bits per
second. This is the max. number of bits that
can be transmitted per second with a
probability of error close to zero.
Shannon’s and Nyquist Theorem…
• If there is no noise in the channel, N=0
• c  .
• The Nyquist Theorem, also known as the sampling theorem, is
a principle that engineers follow in the digitization
of analog signals. For analog-to-digital conversion (ADC) to
result in a faithful reproduction of the signal, slices,
called samples, of the analog waveform must be taken
frequently. The number of samples per second is called the
sampling rate or sampling frequency.
Nyquist Theorem…
• Any analog signal, consists of components at various
frequencies.
• The simplest case is the sine wave, in which all the signal
energy is concentrated at one frequency.
• In practice, analog signals usually have complex waveforms,
with components at many frequencies.
• The highest frequency component in an analog signal
determines the bandwidth of that signal. The higher the
frequency, the greater the bandwidth.
Nyquist Theorem…
• Suppose the highest frequency component, in hertz, for a
given analog signal is fmax.
• According to the Nyquist Theorem, the sampling rate must be
at least 2fmax, or twice the highest analog frequency
component.
• If the sampling rate is less than 2fmax, some of the highest
frequency components in the analog input signal will not be
correctly represented in the digitized output. When such a
digital signal is converted back to analog form by a digital-toanalog converter, false frequency components appear that
were not in the original analog signal. This undesirable
condition is a form of distortion called aliasing.
Examples of Aliasing
• 1. Consider two sinusoidal signals:-
•
s1 (t )  cos 2 (10)t
s 2 (t )  cos 2 (50)t
• Which are sampled at a rate Fs  40hz The corresponding
discrete signals are
•
•
10

) n  cos n
40
2
50
5
s 2 ( n)  cos 2 ( ) n  cos
n
40
2
s1 ( n)  cos 2 (
Example of Aliasing …
• Result is that after digitizing, the value of
signal s2 (5 integer multiple i.e. cos ∏/2)is
identical with s1 (cos∏/2) . In other words
frequency of signal s2 is alias with the
frequency of signal s1.
• So, as per Nyquist criteria, sampling rate Fs
>2fmax of the signal components. In this case Fs
should be 100hz. i.e. twice of 50hz.
Examples of Aliasing…
• 2. Consider the analog signal:• s(t )  3 cos 50t  10 sin 300t  cos100t
• What is the Nyquist rate for this signal?
• Solution; F1,F2,F3 are 25,150,50hz. Respectively.
• (Note If the signal for e.g. is cos2∏(10)t, no need to divide it
by 2. Because it is already in the form of 2∏f. In the other
case, for e.g. cos 50∏t , in order to make it in the form of
2∏f, we have to divide frequency 50hz by 2.
Aliasing Examples…
• Thus fmax is 150hz, therefore Fs >2fmax =300hz
Basic Filter Types
• A filter is a circuit that is designed to pass a specific
band of frequencies, while block all signals outside
this band.
• Application include (but certainly not limited to)
noise rejection, signal separation, smoothing of
digitally generated analog signals, audio signal
shaping etc…
• There are four types of filter; low-pass, high-pass,
band-pass, and band elimination also known as
notch filter.
Basic filter types …
• A low-pass filter is an electronic filter that
passes low
frequency signals but attenuates (reduces
the amplitude of) signals with frequencies
higher than the cutoff frequency Fig.1.
• A high-pass filter (HPF) is a device that passes
high frequencies and attenuates (i.e., reduces
the amplitude of) frequencies lower than
its cutoff frequency Fig.1.
Basic filter types…
• A band-pass filter is a device that
passes frequencies within a certain range and
rejects (attenuates) frequencies outside that
range Fig.2.
• A Notch filter is a filter that passes all
frequencies except those in a stop band
centered on a center frequency Fig.2.
• Notch filters are used to reject unwanted
signals for e.g. spikes in sensitive instruments.
Fig.1. Basic Filter types…
Low Pass filter
High pass filter
Fig.2.Basic Filter types…
Band pass filter
Notch filter
Week 3 and 4
Fourier Series…
Fourier Transform…
Time & Frequency domain concept
• Time-domain graph shows how a signal changes over time
• In the freq. domain, all the component s (freq. and
amplitude) of a sine wave can be represented by a single
vertical line. Similarly, freq. and phase, as shown in Fig. 4
• Frequency domain is a term used to describe the domain for
analysis of signals with respect to frequency, rather than
time.
• Frequency-domain graph shows how much of the signal lies
within each given frequency band over a range of
frequencies. A frequency-domain representation can also
include information on the phase shift.
Time & Frequency domain concept….
• Multiple sine wave called a composite signal, which
is used in communication.
• Example
• S(t) =1 sine10(10000∏t+ ø)
• Where ‘1’ is the amplitude in volts, ’10’ is the no. of
cycles, (angular frequency) ω = 2∏f , so f = 50khz, ‘t’
is the time and ‘ø’ is the phase, in this case it is 0,
because sine wave starts at 0 origin.
• Graphical rep. of this equation is shown in fig.4.
Fig. 4 Time & Frequency domain example
10 periods of 1 Volt, 50KHz
sine wave
Amplitude in the Single Sided
DFT in the 50KHz bin is 1.0 V
Time & Frequency domain concept…
• A given function or signal can be converted
between the time and frequency domains with a
pair of mathematical operators called a transform.
• An example is the Fourier transform, which
decomposes a function into the sum of a
(potentially infinite) number of sine wave frequency
components.
• The 'spectrum' of frequency components is the
frequency domain representation of the signal. The
inverse Fourier transform converts the frequency
domain function back to a time function.
Fourier Analysis
• Fourier analysis provides us a way to view time
domain signal in frequency domain. Fourier series
and Fourier Transform are part of it. The former
deal with periodic signal, which can be represented
as a sum of sinusoids as shown in fig Fig. 5. Actually,
this figure shows the verification of conversion of
square wave function into Fourier series. While the
latter is related with non-periodic signals. This is
shown in Fig.6.
Fourier Analysis …
• Signals like pulse and transient are of practical importance in
communications. These signals cannot be analyzed by
Fourier series due to mathematical constraints.
• The Fourier Transform decomposes a waveform - basically
any real world waveform, into sinusoids. That is, the Fourier
Transform gives us another way to represent a waveform.
• The Fourier series is named in honor of Joseph Fourier
(1768–1830), who made important contributions to the
study of trigonometric series, after preliminary
investigations by Leonhard Euler, Jean le Rond d Alembert,
and Daniel Bernoulli
Fig. 5 Fourier series approximations for a square wave.
Fourier Series
• In mathematics, a Fourier
series decomposes periodic functions or
periodic signals into the sum of a (possibly
infinite) set of simple oscillating functions,
namely sines and cosines (or complex
exponentials).
• Therefore, a signal g(t) can be expressed by a
trigonometric Fourier series over any interval
of duration T0 second as
Trigonometric Fourier Series…
g (t )  a0  a1 cos 0t  a2 cos 20t  ...
...b1 sin 0t  b2 sin 20t

g (t )  a0   an cos n0t  bn sin n0t
n 1
Trigonometric Fourier Series…
• Where 0  2 / T0 , for the coefficients a0,an and
bn we have
1

T0
t1 T0
2

T0
t1 T0
2
bn 
T0
t1 T0
a0
an
 g (t ) dt ,
t1
 g (t ) cos n tdt ,
0
t1
 g (t ) sin n tdt
0
t1
Compact Trigonometric Fourier Series
• The trigonometric Fourier series contains sine
and cosine of the same frequency. We can
combine the two terms in a single term of the
same frequency using the trigonometric
identity:-
Compact Trigonometric Fourier Series…
an cos n0 t  bn sin n0 t  cn cos(n0 t   n )
where,
Cn  a n  b
2
2
n
 bn
 n  tan (
)
an
1
Fourier Transform
• The Fourier transform is a mathematical operation with many
applications in physics and engineering that expresses a
mathematical function of time as a function of frequency,
known as its frequency spectrum; The Fourier Transform is
used for non-periodic or Aperiodic signals as shown in Fig.6.
• The function of time is often called the time
domain representation, and the frequency spectrum
the frequency domain representation. The inverse Fourier
transform expresses a frequency domain function in the time
domain. Each value of the function is usually expressed as a
complex number (called complex amplitude) that can be
interpreted as a magnitude and a phase component.
Fourier Transform…
Fig. 6;Aperiodic signal
Fourier Transform…
• The representation of non-periodic signals by
eternal exponential (everlasting or endless)
can be accomplished by a simple limiting
process. In other words non-periodic signals
can be expressed as a continuous sum
(integral) of a signal, which have no duration.
• This is shown in Fig.7a & 7b.
Figure 7a&b; Aperiodic signal and its
representation as a continuous sum.
Fourier Transform…
• As shown in figure 7b a new periodic signal gp
(t)consisting of signal g(t) repeating itself
every T0 sec. This period is made long enough
so that there is no overlap between the
repeating pulses.
• This new signal gp (t) is a periodic signal and
can be represented by Fourier series.
• (Note derivation is given in the B.P. Lathi book)
Fourier Transform…
• The Fourier transform and its inverse of Aperiodic signal in
terms of eternal exponential function is given below
respectively:-

G ( )   g (t )e
 jwt
d


1
jwt
g (t ) 
G ( )e dt

2 
Fourier Transform…
• Where G(ω) is the frequency domain
representation of the continuous signal g(t),
with frequencies lying in the interval (∞<ω<∞).
• While g(t) is the time domain representation
of the G(ω).
Power Spectral Density
• Power of the periodic signal is distributed among the
various frequency components.
• Power spectral density function (PSD) shows the
strength of the variations(energy) as a function of
frequency. In other words, it shows at which
frequencies variations are strong and at which
frequencies variations are weak.
• The unit of PSD is energy per frequency(width) and
you can obtain energy within a specific frequency
range by integrating PSD within that frequency
range.
Power Spectral Density…
• Computation of PSD is done directly by the
method called FFT or computing
autocorrelation function and then
transforming it.
Power Spectral Density…
• We define the power spectral density (PSD)
(ω) as
2
lim | GT ( ) |
S g ( ) 
d
T 
T
• Where GT(ω), is signal spectrum and ‘T’ is the
time.
Parseval’s Theorem
• Signal Energy in time domain can be related to
the signal spectrum:

1
2
Eg   g (t )d t 
| G( ) | d

2 

2
Properties of Fourier Transform
• Time-Frequency duality
• Symmetry property
– g(t)
G(ω)
– Then G(t) 2∏g(-ω)
• Scaling property
– If g(t)
G(ω)
– Then g(at)
1
 
G

|a|  a 
Properties of Fourier Transform…
• Reciprocity of signal Duration and its
Bandwidth. This suggests that the bandwidth
of a signal is inversely proportional to the
signal duration or width (in seconds)
• Time shifting property
– g(t) G(ω)
 jt 0
– g(t-t0)
G ( )e
Properties of Fourier Transform…
• Frequency-Shifting Property
– g(t)
G(ω);
g (t )e 
 G (  0 )
jwt0
Channel & their types with
distortions
• The channel is a medium – such as wire, coaxial cable a wave guide, an optical fiber, or a
radio link – through which the transmitter
output is sent.
• Distortion; spreading or dispersion of the
pulse will occur if either the amplitude
response or the phase response (linear
distortion) or both are not ideal. For example
in TDM, pulse spreading causes interference.
Channel & their types with
distortions…
• Please note that linear distortion is valid only
for small signals. For large amplitudes, nonlinearity cannot be ignored for example
memory less channel.
Convolution
• The convolution of two signals or functions
g(t) and w(t), denoted as follows:– g(t) * w(t) =

 g ( ) w(t   )d

• Application is in digital filtering