Lecture 5

CENG 3310: Communication Systems

Ref: Text , equations and some figures have been taken from the book (textbook) “Modern Digital and Analog
Communication Systems” by B. P. Lathi and Z. Ding.

Note: Do NOT share these lecture slides to others due to copyright restrictions from the publisher.

CENG 3310 1
Contents
• To introduce linear systems
• To introduce convolution
• Signal Transmission through a Linear System
• Signal distortion during Transmission
• Examples of ideal and real Filters

CENG 3310 2
Linear System
• A system is a black box that converts an input signal 𝑔(𝑡) in an output signal
𝑦(𝑡).

g(t) System y(t)

Linear Time
Invariant
𝑔1 (𝑡) --→ 𝑦1 (𝑡)
𝑔2 (𝑡) -→ 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 ).

CENG 3310 3
Linear 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.

𝜹(𝒕)

• 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 = න ℎ 𝜏 𝑥 𝑡 − 𝜏 𝑑𝜏
−∞

CENG 3310 4
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 𝑒 𝑗𝜃𝑦 (𝑤) = 𝐻 𝑤 𝑒 𝑗𝜃ℎ (𝑤) 𝑋 𝑤 𝑒 𝑗𝜃𝑥 (𝑤)
𝑌 𝑤 𝑒 𝑗𝜃𝑦 (𝑤) = 𝐻 𝑤 𝑋 𝑤 𝑒 𝑗[𝜃ℎ 𝑤 +𝜃𝑥 (𝑤)]
So,
𝑌 𝑤 = 𝐻 𝑤 𝑋 𝑤
𝜃𝑦 𝑤 = 𝜃ℎ 𝑤 + 𝜃𝑥 (𝑤)
CENG 3310 5
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
CENG 3310 6
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.

CENG 3310 7
Ideal and Practical Filters

• Filter: An electronic device or mathematical algorithm to modify the signals.

• In communications, filters are used for separating an information carrying

signal from unwanted corruptions such as interference, noise and distortion
products.
➢Low-pass filter (LPF)
➢High-pass filter (HPF)
➢Bandpass filter (BPF)
➢Bandstop filter (BSF)

CENG 3310 8
Ideal and Practical Filters

• Ideal filters allow distortionless transmission of a certain band of frequencies

and suppression of all the remaining frequencies.

• For simplicity, we often use ideal filter in our deduction, which has a sharp
stop band in frequency domain, and accurate bandwidth.

CENG 3310 9
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

𝑓
𝐻 𝑓 =∏ 𝑒 −𝑗2𝜋𝑓𝑡𝑑
2𝐵
• The ideal low pass filter impulse response will be
−1
𝑓
ℎ 𝑡 =ℱ ∏ 𝑒 −𝑗2𝜋𝑓𝑡𝑑
2𝐵
= 2𝐵 sin𝑐 2𝜋𝐵(𝑡 − 𝑡𝑑 )
CENG 3310 10
Ideal High-Pass and Band-Pass filters

High Pass Filter

Band Pass Filter

CENG 3310 11
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.

CENG 3310 12
Example of a linear system: RC circuit

1ൗ
𝑗𝑤𝐶 1 𝑎
𝐻 𝑤 = = =
1
𝑅 + ൗ𝑗𝑤𝐶 1 + 𝑗𝑤𝑅𝐶 𝑎 + 𝑗𝑤

where,
1
𝑎=
𝑅𝐶
and,
𝑎
𝐻(𝑤) = ⇒ 𝐻(0) = 1, lim 𝐻(𝑤) = 0
𝑎2 + 𝑤2 𝑤
𝑤→∞
𝜃ℎ 𝑤 = − tan−1
𝑎
• Therefore, the circuit behaves as a low-pass filter.
CENG 3310 13
Signal Distortion over a
Communication Channel

• Linear Distortion

• Non-Linear Distortion

• Distortion caused by multipath effects

• Fading channels

CENG 3310 14
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.
Channel transforms/distorts the signal
CENG 3310 15
TDM and FDM

•TDM divides the channel into time slot.

•FDM divides the channel into frequency slots.

16
Example

• A low pass filter transfer function 𝐻(𝑓) is given by

(1 + 𝑘 cos 2𝜋𝑓𝑇)𝑒 −2𝜋𝑓𝑡𝑑 𝑓 <𝐵
𝐻 𝑓 =൝
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).

CENG 3310 17
18
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 (non-linear
equation) expanded in Maclaurin series
𝑦=𝑓 𝑔
𝑦 𝑡 = 𝑎0 + 𝑎1 𝑔 𝑡 + 𝑎2 𝑔2 𝑡 + 𝑎3 𝑔3 𝑡 + ⋯ + 𝑎𝑘 𝑔𝑘 𝑡 + ⋯
• In broadcast communication, high power amplifiers are desirable, but they are non-linear.
• 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.

CENG 3310 19
Example

The input 𝑥(𝑡) and the output 𝑦(𝑡) of a certain nonlinear channel are related
as
𝑦 𝑡 = 𝑥 𝑡 + 0.000158𝑥 2 (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
CENG 3310 20
Example (Contd)

• Verify that the bandwidth of the output signal is twice that of the input signal.

CENG 3310 21
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.

CENG 3310 22
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.

CENG 3310 23
2 Look: Fourier Transform Table
nd
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

CENG 3310 25
Signal Energy: Parseval’s Theorem

• Consider an energy signal 𝑔(𝑡), Parseval’s Theorem states that

∞ ∞
1
𝐸𝑔 = න 𝑔(𝑡) 2 𝑑𝑡 = න 𝐺(𝑤) 2 𝑑𝑤
−∞ 2𝜋 −∞
Proof:

CENG 3310 26
Example
• Consider the signal 𝑔 𝑡 = 𝑒 −𝑎𝑡 𝑢 𝑡 𝑎>0
• Its energy is
∞ ∞
1
𝐸𝑔 = න 𝑔2
𝑡 𝑑𝑡 = න =𝑒 −2𝑎𝑡 𝑑𝑡
−∞ 0 2𝑎
• We now determine 𝐸𝑔 using the signal spectrum 𝐺(𝑤) given
by
1
𝐺 𝑤 =
𝑗𝑤 + 𝑎
• It follows

• Which verifies Parseval’s theorem.

CENG 3310
27
 the distribution of a signal’s energy
over the frequency domain.
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).

CENG 3310 28
Energy Spectral Density (continued)

• The energy spectral density (ESD) 𝜓(𝑤) is thus defined as

𝜓𝑔 𝑓 = 𝐺(𝑓) 2

and
∞
𝐸𝑔 = න 𝜓𝑔 𝑓 𝑑𝑓
−∞

Thus, the ESD of the signal 𝑔 𝑡 = 𝑒 −𝑎𝑡 𝑢(𝑡) of the previous example is
2
1
𝜓𝑔 𝑓 = 𝐺(𝑓) =
(2𝜋𝑓)2 +𝑎2

CENG 3310 29
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.

CENG 3310 30
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

∞ ∞
1
𝐸𝑔 = න 𝑔2 𝑡 𝑑𝑡 = න 𝑒 −2𝑎𝑡 𝑑𝑡 =
−∞ 0 2𝑎

CENG 3310 31
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 2𝜋𝑓0 𝑡
and the spectrum (Fourier Transform) of 𝜑(𝑡) is
1
𝜑 𝑓 = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 )
2
• The ESD of the modulated signal 𝜑(𝑡) is 𝜑(𝑓) 2 , that is
1
𝜓𝜑 𝑓 = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 ) 2
4
• If 𝑓0 ≥ 𝐵, then 𝐺 𝑓 + 𝑓0 and 𝐺 𝑓 − 𝑓0 are non-overlapping, and
1
𝜓𝜑 𝑓 = 𝐺(𝑓 + 𝑓0 ) 2 + 𝐺(𝑓 − 𝑓0 ) 2
4
1 1
= 𝜓𝑔 𝑓 + 𝑓0 + 𝜓𝑔 𝑓 − 𝑓0
4 4

CENG 3310 32
Energy of Modulated Signals
(Contd)

CENG 3310 33
Energy of Modulated Signals
(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 𝑤0 𝑡 is half the energy of
𝑔(𝑡). That is,
1
𝐸𝜑 = 𝐸𝑔
2
• The same applies to power signals. That is, if 𝑔(𝑡) is a power signal then
1
𝑃𝜑 = 𝑃𝑔
2

CENG 3310 34
 Power Spectral Density (PSD)
First consider this:

This is PSD.

Because PSD is always real

and even.