Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 4-9)

30820-Communication Systems
Week 2-3 – Lecture 4-9
(Ref: Chapter 2 of text book)
SIGNALS AND SIGNAL SPACE

Contents
• Signals
• Classification of Signals
• Signal Comparison: Correlation, Cross Correlation Function
• Auto Correlation Function
• Orthogonal Signal Set
• Review of Fourier Series and Fourier Transform and their Properties
308201- Communication Systems 2

Signals
• A signal is a set of information or data.
• Examples
– a telephone or television signal,
– monthly sales of a corporation,
– the daily closing prices of a stock market.
• Signal is a function of a dependent variable against an
independent one.
– For our present course the independent variable is time.
• We deal exclusively with signals that are functions of time.

Classifications of Signals
• Continuous-time and discrete-time signals
• Analog and digital signals
• Periodic and aperiodic signals
• Energy and power signals
• Deterministic and probabilistic signals
• Other Classifications
– even / odd signals,
– One-dimensional / Multi-dimensional

Continuous-time and discrete-time
Signals
• A signal that is specified for every value of time t is a
continuous-time signal.
• A signal that is specified only at discrete values of t is a
discrete-time signals.

Continuous-time and discrete-time
Signals (continued)
• A discrete-time signal can be obtained by sampling a
continuous-time signal.
• In some cases, it is possible to 'undo' the sampling operation.
That is, it is possible to get back the continuous-time signal
from the discrete-time signal.
• How?
• Sampling Theorem
– The sampling theorem states that if the highest frequency in the signal
spectrum is 𝐵, the signal can be reconstructed from its samples taken
at a rate not less than 2𝐵 sample per second.

Analog and digital signals
• A signal whose amplitude can take
on any value in a continuous range is
an analog signal.
• A signal whose amplitude can take a
finite number of values is a digital
signal.
• The concept of analog and digital
signals is different from the concept
of continuous-time and discrete-
time signals.
• For example, we can have a digital
and continuous-time signal, or a
analog and discrete-time signal.

Periodic and Aperiodic Signals
• A signal 𝑔(𝑡) is said to be periodic if for some positive
constant 𝑇0,
𝑔 𝑡 = 𝑔(𝑡 + 𝑇0) for all 𝑡
• A signal is aperiodic if it is not periodic.
• Some famous periodic signals:
– sin 𝑤0𝑡
– cos 𝑤0𝑡
– 𝑒𝑗𝑤0𝑡 = cos 𝑤0𝑡 + j sin 𝑤0𝑡
• where 𝑤0 = 2𝜋
𝑇0
and 𝑇0 is the period of the function.

Periodic Signal
• A periodic signal 𝑔(𝑡) can be generated by periodic extension
of any segment of 𝑔(𝑡) of duration 𝑇0.

Energy and Power signal
• First, define energy.
– The signal energy 𝐸𝑔 of 𝑔(𝑡) is defined (for a real signal) as
𝐸𝑔 =
−∞
∞
𝑔2 𝑡 𝑑𝑡
– In the case of a complex valued signal 𝑔(𝑡), the energy is given by
𝐸𝑔 =
−∞
∞
𝑔∗ 𝑡 𝑔 𝑡 𝑑𝑡 =
−∞
∞
𝑔 𝑡 2𝑑𝑡
• A necessary condition for energy to be finite is that the
signal amplitude approaches zero as 𝑡 → ∞ otherwise the
above integral will not converge. If this condition doesn’t
hold the signal energy is infinite.
• A signal 𝑔(𝑡) is an energy signal if 𝐸𝑔 < ∞.

Energy and Power signal
• A power signal must have an infinite duration.
• In case of signals with infinite energy (e.g., periodic signals), a
more meaningful measure is the signal power.
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔2
(𝑡)𝑑𝑡
• For a complex signal 𝑔(𝑡), the signal power is given by
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔(𝑡) 2𝑑𝑡
• The square root of 𝑃
𝑔 is known as the rms value of 𝑔(𝑡)
• Note: A signal cannot be an energy and a power signal at the
same time.

Energy Signal
Example
• Determine the energy of the following signal.
• Signal Energy calculation
𝐸𝑔 =
−∞
∞
𝑔2 𝑡 𝑑𝑡 =
−1
0
(2)2𝑑𝑡 +
0
∞
(2𝑒
−𝑡
2)2𝑑𝑡 = 4 + 4 = 8

Power Signal
Example
• Why it is a power signal?
– Signal does not approach 0 as 𝑡 → ∞
– Signal is periodic
• Signal power calculation?
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔(𝑡) 2
𝑑𝑡 =
1
2 −1
1
𝑡2
𝑑𝑡 =
1
3
• Its rms value will be?
– Square root of 𝑃
𝑔 i.e., 1
3

Practice Problems
• Find the energy of the following signal.
• Find a suitable measure of the following signal.

Deterministic and Probabilistic Signals
• A signal whose physical description is known completely is a
deterministic signal.
– Mathematical form
– Graphical form
• A signal known only in terms of probabilistic descriptions is a
random signal.
– Mean value
– Mean square value
– Distributions
– Example: Noise, message signals

Other Classifications
• Even and Odd Signals
– A signal is even if 𝑥(𝑡) = 𝑥(−𝑡).
– A signal is odd if 𝑥(𝑡) = −𝑥(−𝑡)
• Examples:
– cos(𝑡) is an even signal.
– sin(𝑡) is an odd signal.
• One-dimensional & Multi-dimensional
– Speech varies as a function of time  one-dimensional
– Image intensity varies as a function of (𝑥, 𝑦) coordinates  multi-
dimensional
One dimensional
Two dimensional

Unit Impulse Function
• The unit impulse function or Dirac delta function is defined as
𝛿 𝑡 = 0 𝑡 ≠ 0
−∞
∞
𝛿 𝑡 𝑑𝑡 = 1
• Can be visualized as a tall, narrow, rectangular pulse of unit area.
• Multiplication of a function by an impulse – Sifting Property
𝑔 𝑡 𝛿 𝑡 = 𝑔 0 𝛿 𝑡
𝑔 𝑡 𝛿 𝑡 − 𝑇 = 𝑔 𝑇 𝛿 𝑡 − 𝑇
−∞
∞
𝑔(𝑡)𝛿 𝑡 − 𝑇 𝑑𝑡 = 𝑔(𝑇)

Unit step function
• Another useful signal is the unit step function 𝑢(𝑡), defined by
𝑢 𝑡 =
1 𝑡 ≥ 0
0 𝑡 < 0
• Observe that
−∞
𝑡
𝛿 𝜏 𝑑𝜏 =
1 𝑡 ≥ 0
0 𝑡 < 0
= 𝑢(𝑡)
• Which also results into
𝑑𝑢
𝑑𝑡
= 𝛿(𝑡)

Sinusoids
• Consider the sinusoid
𝑥 𝑡 = 𝐶 cos(2𝜋𝑓0𝑡 + 𝜃)
• 𝑓0 (measured in Hertz) is the frequency of the sinusoid and
𝑇0 = 1
𝑓0
is the period. 𝐶 is the amplitude and 𝜃 is the phase.
• Sometimes we use 𝑤0 (radians per second) to express 2𝜋𝑓0.
• Important Identities
𝑒±𝑗𝑥 = cos 𝑥 ± 𝑗 sin 𝑥
cos 𝑥 =
1
2
𝑒𝑗𝑥 + 𝑒−𝑗𝑥
sin 𝑥 =
1
2𝑗
𝑒𝑗𝑥 − 𝑒−𝑗𝑥

Signal Operations
Time Shifting Time Scaling Time Reversal or inversion
φ(𝑡) = g(𝑡 − 𝑇) represents “Delay” φ(𝑡) = g(2𝑡) represents
“Compression”
φ 𝑡 = g −𝑡
φ(𝑡) = g(𝑡 + 𝑇) represents “Advance” φ(𝑡) = g(𝑡/2) represents
“Expansion”
Mirror image on vertical axis i.e., y-axis
Note: −𝑥 𝑡 means mirror image on
horizontal axis i.e., x-axis

Signals and Vectors
• Signals and vectors are closely related. For
example,
– A vector has components,
– A signal has also its components.
• g is a certain vector.
– It is specified by its magnitude or length g and
direction.
• Consider a second vector x.
• Different ways to express g in term of vector x
– g = 𝑐x + 𝑒 = 𝑐1x + 𝑒1 = 𝑐2x + 𝑒2
– e is the error vector
– c is the magnitude of projection of g on x
– Choose c to minimize e i.e., e = g − cx

Inner product in vector spaces
• For convenience we define the dot (inner or scalar) product of two
vectors as
g, x = g x cos θ
• Therefore, x, x = x 2
• The length of the component of g along x is g cos 𝜃, but it is also
c 𝑥 .
c 𝑥 = g cos 𝜃
• Multiplying both sides by 𝑥
c 𝑥 2
= g x cos 𝜃 = g, x
c =
g, x
x, x
=
g. x
x 2
• When g, x = 0, we say that g and x are orthogonal to each other
i.e., geometrically, 𝜃 =
𝜋
2

Signals as vectors
• The same notion of inner product can be applied for signals.
• What is the useful part of this analogy?
– We can use some geometrical interpretation of vectors to understand
signals.
• Consider two (energy) signals 𝑦(𝑡) and 𝑥(𝑡).
• The inner product is defined by
𝑦(𝑡), 𝑥(𝑡) =
−∞
∞
𝑦 𝑡 𝑥 𝑡 𝑑𝑡
• For complex signals
𝑦(𝑡), 𝑥(𝑡) =
−∞
∞
𝑦 𝑡 𝑥∗ 𝑡 𝑑𝑡
• The two signals are orthogonal if
𝑦(𝑡), 𝑥(𝑡) = 0

Signals as vectors
• If a signal g(t) is approximated in terms of another real signal
x(t) over an interval (t1, t2), then the best approximation
would be the one that minimizes the size of error e(t) (Error
energy).
• Minimizing error energy would mean putting
• Which would eventually give
• which is optimum value of c that minimizes the energy of the
error signal in the approximation g(t)~cx(t).
2 2
1 1
2 2
( ) [ ( ) ( )]
t t
e
t t
E e t dt g t cx t dt
  
 
0
e
dE
dc

2
2
1
2
1
1
2
( ) ( )
1
( ) ( )
( )
t
t
t
t
x t
t
g t x t dt
c g t x t dt
E
x t dt
 




Signals as vectors
Example
For a square signal g(t) shown below find the component in g(t) of the form sin 𝑡.
𝐠(𝐭) ~ 𝐜 𝐬𝐢𝐧 𝐭
Objective: Select c so that the energy of the error signal is minimum. As we already know that
this condition holds for
c =
1
Ex t1
t2
g t x t dt
Let, x(t) = sin t
So, Ex = 0
2π
sin2
t dt = π
c =
1
𝜋 t1
t2
g t sin 𝑡 dt =
1
𝜋 0
π
sin 𝑡 dt +
1
𝜋 π
2𝜋
− sin 𝑡 dt =
4
𝜋
Therefor, g(t) ~
4
𝜋
sin t

Energy of orthogonal signals
• If vectors 𝑥 and 𝑦 are orthogonal, and if 𝑧 = 𝑥 + 𝑦
𝑧 2 = 𝑥 2+ 𝑦 2 (Pythagorean Theorem)
• If signals 𝑥(𝑡) and 𝑦(𝑡) are orthogonal and if 𝑧(𝑡) = 𝑥(𝑡) + 𝑦(𝑡) then
𝐸𝑧 = 𝐸𝑥 + 𝐸𝑦
• Proof?

Power of orthogonal signals
• The same concepts of orthogonality and inner product extend
to power signals.
• For example,
• 𝑔 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 = 𝐶1 cos(𝑤1𝑡 + 𝜃1) + 𝐶2 cos(𝑤2𝑡 + 𝜃2)
and 𝑤1 ≠ 𝑤2
𝑃𝑥 =
𝐶1
2
2
, 𝑃𝑦 =
𝐶2
2
2
• The signal 𝑥(𝑡) and 𝑦(𝑡) are orthogonal: 𝑦(𝑡), 𝑥(𝑡) = 0.
Therefore,
𝑃
𝑔 = 𝑃𝑥 + 𝑃𝑦 =
𝐶1
2
2
+
𝐶2
2
2

Signal Comparison: Correlation
• Why bother poor undergraduate students with correlation?
– Correlation is widely used in engineering.
• To design receivers in many communication systems
• To identify signals in radar systems
• For classifications.

Signal Comparison: Correlation
• If vectors 𝑥 and 𝑦 are given, we have the correlation measure as
𝑐𝑛 = cos 𝜃 =
𝑦, 𝑥
𝑦 𝑥
• Clearly, −1 ≤ 𝑐𝑛 ≤ 1 and is called correlation coefficient.
• In the case of energy signals:
𝑐𝑛 =
1
𝐸𝑦𝐸𝑥 −∞
∞
𝑦 𝑡 𝑥(𝑡) 𝑑𝑡
• Again, −1 ≤ 𝑐𝑛 ≤ 1
• Auto-Correlation
𝜓𝑔 𝜏 =
−∞
∞
𝑔 𝑡 𝑔(𝑡 + 𝜏) 𝑑𝑡
• It is a measure of the similarity of a signal with its displaced version.

Best friends, worst enemies and
complete strangers
• 𝑐𝑛 = 1. Best friends. This happens when g t = K x(t) and K
is positive.
– The signals are aligned, maximum similarity.
• 𝑐𝑛 = −1. Worst Enemies. This happens when g t = K x(t)
and K is negative.
– The signals are again aligned, but in opposite directions.
– The signals understand each other, but they do not like each other.
• 𝑐𝑛 = 0. Complete Strangers. The two signals are orthogonal.
– We may view orthogonal signals as unrelated signals.

Correlation Examples
• Find the correlation coefficients between:
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤1𝑡)
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1cos(𝑤1𝑡) and 𝑤0 ≠ 𝑤1
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1cos(𝑤0𝑡)
• 𝑐𝑥,𝑦 = 1
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤1𝑡) and 𝑤0 ≠ 𝑤1
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤0𝑡)
• 𝑐𝑥,𝑦 = 1
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦 𝑡 = −𝐴1 sin 𝑤0𝑡
• 𝑐𝑥,𝑦 = −1

Signal representation by
orthogonal signal sets
• Examine a way of representing a signal as a sum of orthogonal
signals.
• We know that a vector can be represented as the sum of
orthogonal vectors.
• Review the case of vectors and extend to signals.

Orthogonal vector space
• Consider a three-dimensional Cartesian vector space
described by three mutually orthogonal vectors, 𝑥1, 𝑥2 and
𝑥3.
𝑥𝑚, 𝑥𝑛 =
0 𝑚 ≠ 𝑛
𝑥𝑚
2 𝑚 = 𝑛
• Any three-dimensional vector can be expressed as a linear
combination of those three vectors:
𝑔 = 𝑐1𝑥1 + 𝑐2𝑥2 + 𝑐3𝑥3
• Where 𝑐𝑖 =
𝑔,𝑥𝑖
𝑥𝑖
2
• In this case, we say that this set of vectors i.e., 𝑥1, 𝑥2, 𝑥3 is a
complete set of orthogonal vectors in 3D space.
• Such vectors are known as a basis vector.

Orthogonal signal space
• Same notions of completeness extend to signals.
• A set of mutually orthogonal signals 𝑥1 𝑡 , 𝑥2 𝑡 , … , 𝑥𝑁(𝑡) is complete if it can
represent any signal belonging to a certain space. For example:
𝑔 𝑡 ≅ 𝑐1𝑥1(𝑡) + 𝑐2𝑥2(𝑡) + ⋯ + 𝑐𝑁𝑥𝑁(𝑡)
≅
𝑛=1
𝑁
𝑐𝑛𝑥𝑛(𝑡)
• If the approximation error is zero for any 𝑔(𝑡) then the set of signals
𝑥1 𝑡 , 𝑥2 𝑡 , … , 𝑥𝑁(𝑡) is complete. In general, the set is complete when 𝑁 → ∞.
• This would lead to the Generalized Fourier series of 𝑔(𝑡) as
𝑔 𝑡 =
𝑛=1
∞
𝑐𝑛𝑥𝑛(𝑡)

Parseval’s Theorem
• We have already established that the energy of the sum of
orthogonal signals is equal to the sum of their energies.
• The energy of 𝑔 𝑡 = 𝑛=1
∞
𝑐𝑛𝑥𝑛(𝑡) can be expressed as the sum of
the energies of the individual orthogonal components
𝐸𝑔 = 𝑐1
2
𝐸1 + 𝑐2
2
𝐸2 + ⋯
=
𝑛=1
∞
𝑐𝑛
2
𝐸𝑛
• In vector space, the square of the length of vector is equal to the
sum of the squares of the lengths of its orthogonal components.
– Parseval’s theorem is the statement of this fact when applied to
signals.

Trigonometric Fourier series
• Consider a signal set
{1, cos 𝑤0𝑡, cos 2𝑤0𝑡, … , cos 𝑛𝑤0𝑡, … , sin 𝑤0𝑡 , sin 2𝑤0𝑡 , … , sin 𝑛𝑤0𝑡}
• A sinusoid of frequency 𝑛𝑤0 is called the 𝑛𝑡ℎ harmonic of the
sinusoid, where 𝑛 is an integer.
• The sinusoid of frequency 𝑤0 is called the fundamental harmonic.
• This set is orthogonal over an interval of duration 𝑇0 = 2𝜋
𝑤0,
which is the period of the fundamental.

• The components of the following set are orthogonal.
{1, cos 𝑤0𝑡, cos 2𝑤0𝑡, … , cos 𝑛𝑤0𝑡, … , sin 𝑤0𝑡 , sin 2𝑤0𝑡 , … , sin 𝑛𝑤0𝑡}
𝑇0
cos 𝑛𝑤0𝑡 cos 𝑚𝑤0𝑡 𝑑𝑡 =
0 𝑚 ≠ 𝑛
𝑇0
2
𝑚 = 𝑛 ≠ 0
𝑇0
sin 𝑛𝑤0𝑡 sin 𝑚𝑤0𝑡 𝑑𝑡 =
0 𝑚 ≠ 𝑛
𝑇0
2
𝑚 = 𝑛 ≠ 0
𝑇0
sin 𝑛𝑤0𝑡 cos 𝑚𝑤0𝑡 𝑑𝑡 = 0 for all 𝑚 and 𝑛
• 𝑇0
means integral over an interval from 𝑡 = 𝑡1 to 𝑡 = 𝑡1 + 𝑇0 for
any value of 𝑡1.

• This set is also complete in 𝑇0. That is, any signal in an interval 𝑡1 ≤
𝑡 ≤ 𝑡1 + 𝑇0can be written as the sum of sinusoids. Or
𝑔 𝑡 = 𝑎0 + 𝑎1 cos 𝑤0𝑡 + 𝑎2 cos 2𝑤0𝑡+…+𝑏1 sin 𝑤0𝑡 + 𝑏2 sin 2𝑤0𝑡 + ⋯
= 𝑎0 +
𝑛=1
∞
𝑎𝑛 cos 𝑛𝑤0𝑡 + 𝑏𝑛 sin 𝑛𝑤0𝑡
• Series coefficients
𝑎𝑛 =
𝑔 𝑡 , cos 𝑛𝑤0𝑡
cos 𝑛𝑤0𝑡 , cos 𝑛𝑤0𝑡
𝑏𝑛 =
𝑔 𝑡 , sin 𝑛𝑤0𝑡
sin 𝑛𝑤0𝑡 , sin 𝑛𝑤0𝑡

Trigonometric Fourier Coefficients
• Therefore
𝑎𝑛 =
𝑡1
𝑡1+𝑇0
𝑔 𝑡 cos 𝑛𝑤0𝑡 𝑑𝑡
𝑡1
𝑡1+𝑇0
cos2𝑛𝑤0𝑡𝑑𝑡
As
𝑡1
𝑡1+𝑇0
cos2𝑛𝑤0𝑡𝑑𝑡 = 𝑇0
2 and 𝑡1
𝑡1+𝑇0
sin2𝑛𝑤0𝑡𝑑𝑡 = 𝑇0
2
We get
𝑎0 =
1
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡)𝑑𝑡
𝑎𝑛 =
2
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡) cos 𝑛𝑤0𝑡 𝑑𝑡 𝑛 = 1,2,3
𝑏𝑛 =
2
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡) sin 𝑛𝑤0𝑡 𝑑𝑡 𝑛 = 1,2,3

Exponential Fourier Series
• Consider a set of exponentials
𝑒𝑗𝑛𝑤0𝑡 𝑛 = 0, ±1, ±2, …
• The components of this set are orthogonal.
• A signal 𝑔 𝑡 can be expressed as an exponential series over
an interval 𝑇0 as follows
𝑔 𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡 𝐷𝑛 =
1
𝑇0 𝑇0
𝑔(𝑡)𝑒−𝑗𝑛𝑤0𝑡 𝑑𝑡

Trigonometric and exponential
Fourier series
• Trigonometric and exponential Fourier series are related. In
fact, a sinusoid in the trigonometric series can be expressed
as a sum of two exponentials using Euler's formula.
𝐶𝑛 cos(𝑛𝑤0𝑡 + 𝜃𝑛) =
𝐶𝑛
2
𝑒𝑗(𝑛𝑤0𝑡+𝜃𝑛) + 𝑒−𝑗(𝑛𝑤0𝑡+𝜃𝑛)
=
𝐶𝑛
2
𝑒𝑗𝜃𝑛 𝑒𝑗𝑛𝑤0𝑡
+
𝐶𝑛
2
𝑒−𝑗𝜃𝑛 𝑒−𝑗𝑛𝑤0𝑡
= 𝐷𝑛𝑒𝑗𝑛𝑤0𝑡 + 𝐷−𝑛𝑒−𝑗𝑛𝑤0𝑡
Where
𝐷𝑛 =
𝐶𝑛
2
𝑒𝑗𝜃𝑛 and 𝐷−𝑛 =
𝐶𝑛
2
𝑒−𝑗𝜃𝑛

Parseval's Theorem
• Exponential Fourier series representation
𝑔 𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡
• Where
𝐷𝑛 =
1
𝑇0 𝑇0
𝑔(𝑡)𝑒−𝑗𝑛𝑤0𝑡
𝑑𝑡
• Power for the exponential representation is
𝑃
𝑔 =
𝑛=−∞
∞
𝐷𝑛
2

Example
• Find the exponential Fourier series for the following signal

Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 4-9)

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