Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 10-15)

30820-Communication Systems
Week 4-5 – Lecture 10-15
(Ref: Chapter 3 of text book)
ANALYSIS AND TRANSMISSION OF SIGNALS

Contents
• Aperiodic Signal Representation by Fourier Integral
• Transforms of Some useful Functions
• Properties of the Fourier Transform
• Signal Transmission Through A Linear System
• Ideal Versus Practical Filters
• Signal Distortion Over a Communication Channel
• Signal Energy and Energy Spectral Density
• Signal Power and Power Spectral Density
308201- Communication Systems 2

Introduction
• We electrical engineers think of signals in terms of their
spectral content.
• We have studied the spectral representation of periodic
signals i.e., Fourier Series
• We now extend this spectral representation to the case of
aperiodic signals.
– Fourier Integrals

Aperiodic signal representation
• Signals encountered in real-life are often non-periodic, and
more importantly, they are samples of random (stochastic)
processes, such as speech signals, imagery, etc.
• The signals we use in communication systems are aperiodic
(non-periodic) in nature. In order to study their behavior in
frequency-domain we follow a simple procedure:
– First we form a periodic extension.
– Next we study the Fourier series for this new signal set.
– Finally, we consider the limiting case when the period is allowed to
become infinity, which is equivalent to saying let the signal become
aperiodic.

Aperiodic signal representation
• We have an aperiodic signal 𝑔(𝑡) and we consider a periodic
version 𝑔𝑇0
(𝑡) of such signal obtained by repeating 𝑔(𝑡) every
𝑇0 seconds.

The periodic signal 𝑔𝑇0
(𝑡)
• The periodic signal 𝑔𝑇0
(𝑡)can be expressed in terms of 𝑔(𝑡) as
follows:
𝑔𝑇0
𝑡 =
𝑛=−∞
∞
𝑔(𝑡 − 𝑛𝑇0)
Notice that, if we let 𝑇0 → ∞, we have that
lim
𝑇0→∞
𝑔𝑇0
𝑡 = 𝑔(𝑡)

The Fourier representation of 𝑔𝑇0
(𝑡)
• The signal 𝑔𝑇0
𝑡 is periodic, so it can be represented in terms of its
Fourier series.
• The basic intuition here is that the Fourier series of 𝑔𝑇0
𝑡 will also
represent 𝑔(𝑡) in the limit for 𝑇0 → ∞.
• The exponential Fourier series of 𝑔𝑇0
𝑡 is
𝑔𝑇0
𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡
Where
𝐷𝑛 =
1
𝑇0 −𝑇0
2
𝑇0
2
𝑔𝑇0
𝑡 𝑒−𝑗𝑛𝑤0𝑡 𝑑𝑡
and
𝑤0 =
2𝜋
𝑇0

The Fourier representation of 𝑔𝑇0
(𝑡)
• Integrating 𝑔𝑇0
𝑡 over (−𝑇0
2 , 𝑇0
2)is the same as integrating
𝑔(𝑡) over (−∞, ∞). So we can write
𝐷𝑛 =
1
𝑇0 −∞
∞
𝑔 𝑡 𝑒−𝑗𝑛𝑤0𝑡 𝑑𝑡
• If we define a function
𝐺 𝑤 =
−∞
∞
𝑔 𝑡 𝑒−𝑗𝑤𝑡 𝑑𝑡
then we can write the Fourier coefficients 𝐷𝑛 as follows
𝐷𝑛 =
1
𝑇0
𝐺(𝑛𝑤0)

Computing the lim
𝑇0→∞
𝑔𝑇0
𝑡
• Thus 𝑔𝑇0
𝑡 can be expressed as
𝑔𝑇0
𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡 =
𝑛=−∞
∞
𝐺(𝑛𝑤0)
𝑇0
𝑒𝑗𝑛𝑤0𝑡
• Assuming
1
𝑇0
=
∆𝑤
2𝜋
, we get
𝑔𝑇0
𝑡 =
𝑛=−∞
∞
𝐺(𝑛∆𝑤)∆𝑤
2𝜋
𝑒𝑗(𝑛∆𝑤)𝑡
• In the limit for 𝑇0 → ∞, ∆𝑤 → 0 and 𝑔𝑇0
𝑡 → 𝑔(𝑡)
• We thus get
𝑔 𝑡 = lim
𝑇0→∞
𝑔𝑇0
𝑡 = lim
∆𝑤→0
𝑛=−∞
∞
𝐺(𝑛∆𝑤)∆𝑤
2𝜋
𝑒𝑗(𝑛∆𝑤)𝑡
=
1
2𝜋 −∞
∞
𝐺(𝑤)𝑒𝑗𝑤𝑡𝑑𝑤

The Fourier series becomes the
Fourier integral in the limit

Fourier Transform and
Inverse Fourier Transform
• The spectral representation 𝐺(𝑤) of 𝑔(𝑡), that is, from
𝐺 𝑤 =
−∞
∞
𝑔(𝑡) 𝑒−𝑗𝑤𝑡
𝑑𝑡
We can obtain 𝑔(𝑡) back by computing
𝑔 𝑡 =
1
2𝜋 −∞
∞
𝐺(𝑤)𝑒𝑗𝑤𝑡 𝑑𝑤
Fourier Transform of 𝒈 𝒕 Conjugate Symmetric Property
𝐺 𝑤 = −∞
∞
𝑔(𝑡) 𝑒−𝑗𝑤𝑡𝑑𝑡 If 𝑔(𝑡) is a real function of 𝑡, then
Inverse Fourier Transform 𝐺 −𝑓 = 𝐺∗
(𝑓)
• 𝑔 𝑡 =
1
2𝜋 −∞
∞
𝐺(𝑤)𝑒𝑗𝑤𝑡
𝑑𝑤 Therefore,
Fourier Transform relationship 𝐺 −𝑓 = 𝐺 𝑓
• 𝑔 𝑡 ⇔ 𝐺(𝑤) 𝜃𝑔 −𝑓 = −𝜃𝑔(𝑓)

Example
• Find the Fourier transform of 𝑔(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡)

• The Unit Gate Function: A square pulse with height 1, and
with unit width, centered at origin is called unite gate
function.
• The unit gate function 𝑟𝑒𝑐𝑡(𝑥) is defined as:
𝑟𝑒𝑐𝑡 𝑥 =
0 𝑥 > 1
2
1 𝑥 < 1
2
Transforms of Some
Useful Functions

Transforms of Some
Useful Functions
• Sinc Function: The function
sin 𝒙
𝒙
is the “sine over argument”
function denoted by sinc(𝑥) .
• sinc(𝑥) is an even function of 𝑥.
• sinc(𝑥) = 0 when sin(𝑥) = 0 and x ≠ 0
• Using L’Hopital’s rule, we find that sinc 0 = 1
• sinc(𝑥) is the product of an oscillating signal sin(𝑥) and a
monotonically decreasing function 1
𝑥

Example
• Find the Fourier transform of 𝑔 𝑡 = 𝑟𝑒𝑐𝑡 𝑡
𝜏 .

Properties of the Fourier Transform
• Fourier Transform Table
– Time-Frequency Duality
• Symmetry of Fourier transformation
• Time and frequency shifting property
• Convolution
• Time differentiation and time integration

Fourier Transform Table

Time-Frequency Duality
* Note: If we consider 𝐺(𝑓) then we don’t consider 1/2π

Symmetry Property
• Consider the Fourier transform pair
𝑔(𝑡) ⇔ 𝐺(𝑤)
Then,
𝐺 𝑡 ⇔ 2π𝑔 −𝑤
Example

Scaling Property
𝑔(𝑡) ⇔ 𝐺(𝑤)
Then,
𝑔 𝑎𝑡 ⇔
1
𝑎
𝐺
𝑤
𝑎
Example

Time-Shifting Property
𝑔(𝑡) ⇔ 𝐺(𝑤)
• Time shifting introduces phase shift
𝑔 𝑡 − 𝑡0 ⇔ 𝐺 𝑤 𝑒−𝑗𝑤𝑡0
Example

Frequency-Shifting Property
𝑔(𝑡) ⇔ 𝐺(𝑤)
• Exponential Multiplication introduces frequency shift
𝑔 𝑡 𝑒𝑗𝑤0𝑡 ⇔ 𝐺 𝑤 − 𝑤0
or
𝑔 𝑡 𝑒−𝑗𝑤0𝑡
⇔ 𝐺 𝑤 + 𝑤0
• Cosine multiplication leads to
𝑔 𝑡 cos 𝑤0𝑡 =
1
2
𝑔 𝑡 𝑒𝑗𝑤0𝑡
+ 𝑔(𝑡)𝑒−𝑗𝑤0𝑡
=
1
2
𝐺(𝑤 − 𝑤0) + 𝐺(𝑤 + 𝑤0)

Frequency-Shifting Property

Convolution
• The convolution of two functions 𝑔(𝑡) and 𝑤(𝑡) is given as
𝑔 𝑡 ∗ 𝑤 𝑡 =
−∞
∞
𝑔 𝜏 𝑤 𝑡 − 𝜏 𝑑𝜏
• Consider two waveforms
𝑔1(𝑡) ⇔ 𝐺1(𝑤) and 𝑔2(𝑡) ⇔ 𝐺2(𝑤)
• Convolution in time domain
𝑔1 𝑡 ∗ 𝑔2(𝑡) ⇔ 𝐺1(𝑤)𝐺2(𝑤)
• Convolution in frequency domain
𝑔1 𝑡 𝑔2 𝑡 ⇔
1
2𝜋
𝐺1 𝑤 ∗ 𝐺2(𝑤)

Time Differentiation and
Time Integration
• Consider the Fourier transform relationship
𝑔(𝑡) ⇔ 𝐺(𝑤)
• The following relationship exists for integration
−∞
𝑡
𝑔 𝜏 𝑑𝜏 ⇔
𝐺(𝑤)
𝑗𝑤
+ 𝜋𝐺 0 𝛿 𝑤
• The following relationship exists for differentiation
𝑑𝑔
𝑑𝑡
⇔ 𝑗𝑤𝐺(𝑤)
𝑑𝑛𝑔
𝑑𝑡𝑛 ⇔ 𝑗𝑤 𝑛
𝐺(𝑤)

Important Fourier Transform Operation

Signal Transmission over
a Linear System
• To introduce linear systems
• To introduce convolution
• To examine signal transmission through a linear system
• To introduce signal distortion during transmission
• To give examples of real and ideal filters

Linear System
• A system is a black box that converts an input signal 𝑔(𝑡) in an
output signal 𝑦(𝑡).
• Assume the output of a signal 𝑔1(𝑡) is 𝑦1(𝑡) and the output of
𝑔2(𝑡) is 𝑦2(𝑡).
– The system is linear if the output of 𝑔1(𝑡) + 𝑔2(𝑡) is
𝑦1(𝑡)+𝑦2(𝑡).
– A system is time invariant if its properties do not change
with the time. That is, if the response to 𝑔(𝑡) is 𝑦(𝑡), then
the response to 𝑔(𝑡 − 𝑡0) is going to be y(𝑡 − 𝑡0).
Linear Time
Invariant System
h(t)
g(t) y(t)

Linear Time Invariant (LTI) System
• Consider a linear time invariant (LTI) system. Assume the input
signal is a Dirac delta function δ(t).
– The output will be the impulse response of the system.
• ℎ(𝑡) is called the “unit impulse response” function.
• With ℎ(𝑡), we can relate the input to its output signal through
the convolution formula:
𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 =
−∞
∞
ℎ 𝜏 𝑥 𝑡 − 𝜏 𝑑𝜏
𝜹(𝒕)

Frequency Response of LTI systems
• If 𝑥 𝑡 ⇔ 𝑋 𝑤 and ℎ(𝑡) ⇔ 𝐻 𝑤 then the convolution reduces to a
product in Fourier domain
𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 ⇔ 𝑌 𝑤 = 𝐻 𝑤 𝑋 𝑤
• 𝐻 𝑤 is called the “system transfer function” or the “system frequency
response” or the “spectral response”.
𝑌 𝑤 = 𝐻 𝑤 𝑋 𝑤
𝑌 𝑤 𝑒𝑗𝜃𝑦(𝑤)
= 𝐻 𝑤 𝑒𝑗𝜃ℎ(𝑤) 𝑋 𝑤 𝑒𝑗𝜃𝑥(𝑤)
𝑌 𝑤 𝑒𝑗𝜃𝑦(𝑤)
= 𝐻 𝑤 𝑋 𝑤 𝑒𝑗[𝜃ℎ 𝑤 +𝜃𝑥(𝑤)]
So,
𝑌 𝑤 = 𝐻 𝑤 𝑋 𝑤
𝜃𝑦 𝑤 = 𝜃ℎ 𝑤 + 𝜃𝑥(𝑤)

Distortionless Transmission
• Transmission is said to be distortionless if the input and the
output have identical wave shapes with a multiplicative
constant.
– A delayed output that retains the input waveform is also
considered distortionless.
• Given an input signal 𝑥(𝑡), the output differs from the input
only by a multiplying constant and a finite time delay
𝑦 𝑡 = 𝑘. 𝑥(𝑡 − 𝑡𝑑)
• The Fourier transform of this equation yields
𝑌 𝑓 = 𝑘𝑋(𝑓)𝑒−𝑗2𝜋𝑓𝑡𝑑

Distortionless Transmission
• As we know that 𝑌 𝑓 = 𝐻 𝑓 𝑋(𝑓)
• The transfer function of a distortionless transmission system is
𝐻 𝑓 = 𝑘𝑒−𝑗2𝜋𝑓𝑡𝑑
We can write,
𝐻 𝑓 = 𝑘
𝜃ℎ 𝑓 = −2𝜋𝑓𝑡𝑑
• The amplitude response 𝐻 𝑓 of a distortionless
transmission system must be a constant and the phase
response 𝜃ℎ 𝑓 must be a linear function of 𝑓 going through
the origin at 𝑓 = 0.

Ideal and Practical Filters
• Filter: An electronic device or mathematical algorithm to
modify the signals.
• In communications, filters are used for separating an
information bearing signal from unwanted contaminations
such as interference, noise and distortion products.
– Low-pass filter (LPF)
– High-pass filter (HPF)
– Bandpass filter (BPF)
– Bandstop filter (BSF)

Ideal and Practical Filters
• Ideal filters allow distortionless transmission of a certain band
of frequencies and suppression of all the remaining
frequencies.
• Practical filters have long tails, complex impulse response,
non-fixed bandwidth, and complex transfer function
expression.
• For simplicity, we often use ideal filter in our deduction.
Which has sharp stop band in frequency domain, and
accurate bandwidth.

Ideal Low Pass Filter
• The ideal low pass filter, allows all components below 𝑓 = 𝐵 𝐻𝑧 to pass
without distortion and suppresses all components above 𝑓 = 𝐵 𝐻𝑧
• The ideal low pass filter response can be expressed as
𝐻 𝑓 = ∏
𝑓
2𝐵
𝑒−𝑗2𝜋𝑓𝑡𝑑
• The ideal low pass filter impulse response will be
ℎ 𝑡 = ℱ−1 ∏
𝑓
2𝐵
𝑒−𝑗2𝜋𝑓𝑡𝑑
= 2𝐵 sin𝑐 2𝜋𝐵(𝑡 − 𝑡𝑑)

Ideal High-Pass and Band-Pass filters
High Pass Filter
Band Pass Filter

Practical Filters
• The filters in the previous examples are ideal filters.
• They are not realizable since their unit impulse responses are
everlasting (think of the sinc function).
• Physically realizable filter impulse response ℎ(𝑡) = 0 for 𝑡 <
0.
• Therefore, we can only obtain approximated version of the
ideal low-pass, high-pass and band-pass filters.

Example of a linear system: RC circuit
𝐻 𝑤 =
1
𝑗𝑤𝐶
𝑅 + 1
𝑗𝑤𝐶
=
1
1 + 𝑗𝑤𝑅𝐶
=
𝑎
𝑎 + 𝑗𝑤
where,
𝑎 =
1
𝑅𝐶
and,
𝐻(𝑤) =
𝑎
𝑎2 + 𝑤2
⇒ 𝐻(0) = 1, lim
𝑤→∞
𝐻(𝑤) = 0
𝜃ℎ 𝑤 = − tan−1
𝑤
𝑎
• Therefore, the circuit behaves as a low-pass filter.

Signal Distortion over a
Communication Channel
• Linear Distortion
• Non-Linear Distortion
• Distortion caused by multipath effects
• Fading channels

Linear Distortion
• Caused due to channel’s non-ideal characteristics of either the
magnitude or phase or both.
• For a time limited pulse, spreading or “dispersion” will occur if
either the amplitude response or the phase response or both
are non ideal.
• For TDM, it causes interference in adjacent channels (cross
talk).
• For FDM, it causes dispersion in each multiplexed signal which
will distort the spectrum of each signal, but no interference,
since each signal occupies a separate channel.

Example
• A low pass filter transfer function 𝐻(𝑓) is given by
𝐻 𝑓 = (1 + 𝑘 cos 2𝜋𝑓𝑇)𝑒−2𝜋𝑓𝑡𝑑 𝑓 < 𝐵
0 𝑓 > 𝐵
A pulse 𝑔(𝑡) band-limited to 𝐵 𝐻𝑧 is applied at the input of the
filter. Find the output 𝑦(𝑡).

Nonlinear Distortion
• Nonlinear distortion is caused by larger signal amplitudes.
• Changes a band limited frequency spectrum 𝐵 𝐻𝑧 to 𝑘𝐵 𝐻𝑧.
• In case of nonlinear channels, input 𝑔 and output 𝑦 are related as a
function expanded in Maclaurin series
𝑦 = 𝑓 𝑔
𝑦 𝑡 = 𝑎0 + 𝑎1𝑔 𝑡 + 𝑎2𝑔2
𝑡 + 𝑎3𝑔3
𝑡 + ⋯ + 𝑎𝑘𝑔𝑘
𝑡 + ⋯
• In broadcast communication, high power amplifiers are desirable, but they
are non-linear.
• Linear distortion causes interference among signals within the same
channel.
• Spectral dispersion due to nonlinear distortion causes interference among
signals using different frequency channels.
– TDM faces no threat from it.
– FDM, faces serious interference problems due to this spectral
dispersion.

Example
The input 𝑥(𝑡) and the output 𝑦(𝑡) of a certain nonlinear
channel are related as
𝑦 𝑡 = 𝑥 𝑡 + 0.000158𝑥2(𝑡)
• Find the output signal 𝑦(𝑡) and its spectrum 𝑌(𝑓) if the input
signal is 𝑥(𝑡) = 2000sinc(2000𝜋𝑡).
Desired Signal
Unwanted Distortion

Example (Contd)
• Verify that the bandwidth of the output signal is twice that of
the input signal.

Distortion due to multipath effects
• In radio links, the signal can be received by direct path
between the transmission and the receiving antenna and also
by reflection from nearby objects.
• Similar behavior observed for ionosphere.

Fading Channels
• Practically channel characteristics vary with time because of
periodic and random changes in the propagation
characteristics of the medium, causing random attenuation of
the signal. Also termed as “fading”
• Can be reduced by “Automatic Gain Control” (AGC).
• Fading may be strongly frequency dependent where different
frequency components are affected unequally.
– Such fading is called frequency-selective fading.
– Multipath propagation can cause frequency-selective fading.

Energy/Power Signals and
Energy/Power Spectral Density
• To introduce Energy spectral density (ESD)
• Input and Output Energy Spectral Densities
• To introduce Power spectral density (PSD)
• Input and Output Power Spectral Densities

Signal Energy: Parseval’s Theorem
• Consider an energy signal 𝑔(𝑡), Parseval’s Theorem states that
𝐸𝑔 =
−∞
∞
𝑔(𝑡) 2𝑑𝑡 =
1
2𝜋 −∞
∞
𝐺(𝑤) 2𝑑𝑤
Proof:

Example
• Consider the signal 𝑔 𝑡 = 𝑒−𝑎𝑡𝑢 𝑡 𝑎 > 0
• Its energy is
𝐸𝑔 =
−∞
∞
𝑔2 𝑡 𝑑𝑡 =
0
∞
𝑒−2𝑎𝑡𝑑𝑡 =
1
2𝑎
• We now determine 𝐸𝑔 using the signal spectrum 𝐺(𝑤) given
by
𝐺 𝑤 =
1
𝑗𝑤 + 𝑎
• It follows
• Which verifies Parseval’s theorem.

Energy Spectral Density
• Parseval’s theorem can be interpreted to mean that the
energy of a signal 𝑔(𝑡) is the result of energies contributed by
all spectral components of a signal 𝑔(𝑡).
• The contribution of a spectral component of frequency 𝑓 is
proportional to 𝐺(𝑓) 2.
• Therefore, we can interpret 𝐺(𝑓) 2 as the energy per unit
bandwidth of the spectral components of 𝑔(𝑡) centered at
frequency 𝑓.
• In other words, 𝐺(𝑓) 2 is the energy spectral density of 𝑔(𝑡).

Energy Spectral Density (continued)
• The energy spectral density (ESD) 𝜓(𝑤) is thus defined as
𝜓𝑔 𝑓 = 𝐺(𝑓) 2
and
𝐸𝑔 =
−∞
∞
𝜓𝑔 𝑓 𝑑𝑓
Thus, the ESD of the signal 𝑔 𝑡 = 𝑒−𝑎𝑡
𝑢(𝑡) of the previous
example is
𝜓𝑔 𝑓 = 𝐺(𝑓) 2 =
1
(2𝜋𝑓)2+𝑎2

Essential Bandwidth of a signal
• The spectra of most signals extend to infinity.
• But since energy of practical signal is finite, signal spectrum →
0, as frequency →∞.
• Most of the signal energy is contained in a certain band of
𝐵 𝐻𝑧, we can suppress the spectrum beyond 𝐵 𝐻𝑧 with little
effect on shape or energy.
• The bandwidth 𝐵 is called the essential bandwidth of the
signal
• The criterion for suppressing 𝐵 depends on the error
tolerance in a particular application
• For example, we may say that select 𝐵 to be that bandwidth
that contains 95% of the signal energy.

Example
• Determine the essential Bandwidth 𝑊 (rad/sec) of the
following signal if the essential band is required to contain
95% of the signal energy.
𝑔 𝑡 = 𝑒−𝑎𝑡
𝑢 𝑡 𝑎 > 0
𝐸𝑔 =
−∞
∞
𝑔2
𝑡 𝑑𝑡 =
0
∞
𝑒−2𝑎𝑡
𝑑𝑡 =
1
2𝑎

Energy of Modulated Signals
• We have seen that modulation shifts the signal spectrum 𝐺(𝑓) to the left and right
by 𝑓0. We now show that a similar thing happens to the ESD of the modulated
signal.
• Let 𝑔(𝑡) be a baseband signal band limited to 𝐵 𝐻𝑧. The amplitude modulated
signal 𝜑(𝑡) is
𝜑(𝑡) = 𝑔(𝑡) cos 2𝜋𝑓0𝑡
and the spectrum (Fourier Transform) of 𝜑(𝑡) is
𝜑 𝑓 =
1
2
𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0)
• The ESD of the modulated signal 𝜑(𝑡) is 𝜑(𝑓) 2, that is
𝜓𝜑 𝑓 =
1
4
𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0) 2
• If 𝑓0 ≥ 𝐵, then 𝐺 𝑓 + 𝑓0 and 𝐺 𝑓 − 𝑓0 are non-overlapping, and
𝜓𝜑 𝑓 =
1
4
𝐺(𝑓 + 𝑓0) 2
+ 𝐺(𝑓 − 𝑓0) 2
=
1
4
𝜓𝑔 𝑓 + 𝑓0 +
1
4
𝜓𝑔 𝑓 − 𝑓0

(Contd)

(Contd)
• Observe that the area under 𝜓𝜑 𝑓 is half the area under
𝜓𝑔 𝑓 because the energy of the signal is proportional to the
area under its ESD.
• The energy of the modulated signal 𝜑(𝑡) = 𝑔(𝑡) cos 𝑤0𝑡 is
half the energy of 𝑔(𝑡). That is,
𝐸𝜑 =
1
2
𝐸𝑔
• The same applies to power signals. That is, if 𝑔(𝑡) is a power
signal then
𝑃𝜑 =
1
2
𝑃
𝑔

Time Autocorrelation Function
and ESD
• For a real signal the autocorrelation function 𝜓𝑔 𝑡 is
𝜓𝑔 𝜏 =
−∞
∞
𝑔 𝑡 𝑔 𝑡 + 𝜏 𝑑𝑡
• Do you remember the correlation of two signals? The
autocorrelation function measure the correlation between
𝑔(𝑡) and all its translated versions.
• Notice
𝜓𝑔 𝜏 = 𝜓𝑔 −𝜏
𝜓𝑔 𝜏 = 𝑔 𝜏 ∗ 𝑔 −𝜏
• Autocorrelation function of 𝑔(𝑡) and its ESD form a Fourier
transform pair i.e.,
𝜓𝑔 𝜏 ⇔ 𝜓𝑔 𝑤 = 𝐺(𝑤) 2

Time Autocorrelation Function
and ESD
• The Fourier transform of the autocorrelation function is the
Energy Spectral Density! i.e.,
𝜓𝑔 𝜏 ⇔ 𝜓 𝑤 = 𝐺(𝑤) 2
Proof:
The Fourier transform of 𝑔(𝑡 + 𝜏) is 𝐺(𝑤)𝑒𝑗𝑤𝑡. Therefore,
• Thus correlation can be viewed as the time domain
counterpart of energy spectral density!

ESD of the Input and the Output
• If 𝑔(𝑡) and 𝑦(𝑡) are the input and the corresponding output of a LTI
system, then
𝑌(𝑤) = 𝐻(𝑤)𝐺(𝑤)
Therefore,
|𝑌(𝑤)|2
= |𝐻(𝑤)|2
|𝐺(𝑤)|2
This shows that
𝜓𝑦 𝑤 = 𝐻 𝑤 2
𝜓𝑔(𝑤)
• Thus, the output signal ESD is |𝐻(𝑤)|2
times the input signal ESD.

• The power 𝑃
𝑔 of a real signal 𝑔(𝑡) is given by
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔2
(𝑡) 𝑑𝑡
• Let 𝑔(𝑡) be a power signal with infinite energy. The truncated signal
𝑔𝑇 𝑡 is an energy signal as 𝑇 is finite.
𝑔𝑇 𝑡 = 𝑔 𝑡 𝑟𝑒𝑐𝑡
𝑡
𝑇
=
𝑔(𝑡) 𝑡 ≤ 𝑇
2
0 𝑡 > 𝑇
2
Signal Power and
Power Spectral Density

Signal Power and
• As long as 𝑇 is finite, the truncated signal has finite energy
𝑃
𝑔 = lim
𝑇→∞
𝐸𝑔𝑇
𝑇
𝐸𝑔𝑇 =
−∞
∞
𝑔𝑇
2 𝑡 𝑑𝑡 =
1
2𝜋 −∞
∞
𝐺𝑇(𝑤) 2𝑑𝑤
• Hence, 𝑃
𝑔, the power of 𝑔(𝑡) is given by
𝑃
𝑔 = lim
𝑇→∞
𝐸𝑔𝑇
𝑇
= lim
𝑇→∞
1
𝑇
1
2𝜋 −∞
∞
𝐺𝑇(𝑤) 2𝑑𝑤
• Exchange the order of integration and the limiting operation.
Thus
𝑃
𝑔 =
1
2𝜋 −∞
∞
lim
𝑇→∞
𝐺𝑇(𝑤) 2
𝑇
𝑑𝑤

Signal Power and
• Let us define
𝑆𝑔 𝑤 = lim
𝑇→∞
𝐺𝑇(𝑤) 2
𝑇
• The frequency dependent function 𝑆𝑔(𝑤) is called the power
Spectral Density (PSD) of 𝑔(𝑡). Thus,
𝑃
𝑔 =
1
2𝜋 −∞
∞
𝑆𝑔(𝑤) 𝑑𝑤
• The power is
1
2𝜋
times the area under the PSD.
• The result is parallel to ESD and signal energy relationship for
energy signals.
• If 𝑔(𝑡) is a voltage signal, the units of PSD are volts squared
per 𝐻𝑧.

Time autocorrelation Function of
Power Signals
• The (time) autocorrelation function 𝑅𝑔(𝜏) of a real deterministic
power signal 𝑔(𝑡) is defined as
𝑅𝑔 𝜏 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔 𝑡 𝑔(𝑡 + 𝜏) 𝑑𝑡
• Using the same derivation as ESD, autocorrelation can be viewed as
the time domain counterpart of power spectral density.
𝑅𝑔 𝜏 ⇔ 𝑆𝑔(𝑤)
• If 𝑔(𝑡) and 𝑦(𝑡) are the input and the corresponding output of a LTI
system, then
𝑆𝑦 𝑤 = 𝐻(𝑤) 2𝑆𝑔(𝑤)
• Thus, the output signal PSD is 𝐻(𝑤) 2
the input signal PSD.

Relationships for Energy
and Power Signals

Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 10-15)

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