signals and systems chapter3-part3_signals and systems chapter3-part3.pdf

Signals and Systems
Chapter 3
FOURIER SERIES REPRESENTATION OF PERIODIC
SIGNALS
Islam K. Sharawneh
islam.Sharawneh@ptuk.edu.ps
Signals and Systems - 12130302 - Islam K. Sharawneh 1
12/6/2023

INTRODUCTION
• The representation and analysis of LTI systems through the convolution sum as developed in Chapter 2 is based on
representing signals as linear combinations of shifted impulses.
• As in Chapter 2, the starting point for our discussion is the development of a representation of signals as linear
combinations of a set of basic signals. For this alternative representation we use complex exponentials. The resulting
representations are known as the continuous-time and discrete-time Fourier series and transform.
• The response of an LTI system to any input consisting of a linear combination of basic signals is the same linear
combination of the individual responses to each of the basic signals.
• As we will find in the current chapter, the response of an LTI system to a complex exponential also has a particularly
simple form, which then provides us with another convenient representation for LTI systems and with another way in which
to analyze these systems and gain insight into their properties.
• In this chapter, we focus on the representation of continuous-time periodic signals referred to as the Fourier series.
• In Chapters 4 (The Continuous-Time Fourier Transform) we extend the analysis to the Fourier transform
representation of broad classes of aperiodic, finite energy signals.
12/6/2023 Signals and Systems - 12130302 - Islam K. Sharawneh 2

INTRODUCTION
• We will see that if the input to an LTI system is
expressed as a linear combination of periodic
complex exponentials or sinusoids, the output can
also be expressed in this form, with coefficients that
are related in a straightforward way to those of the
input.
• Fourier was born on March 21, 1768, in Auxerre,
France, and by the time of his entrance into the
controversy concerning trigonometric series, he had
already had a lifetime of experiences. His many
contributions-in particular, those concerned with the
series and transform that carry his
name-are made even more impressive by the
circumstances under which he worked.
His revolutionary discoveries, although not completely
appreciated during his own lifetime, have had a major
impact on the development of mathematics and have
been and still are of great importance in an extremely
wide range of scientific and engineering disciplines.

INTRODUCTION
There are four distinct Fourier representations :
1. Continuous-Time Fourier Series (CTFS).
2. Discrete-Time Fourier Series (DTFS).
3. Continuous-Time Fourier Transform (CTFT).
4. Discrete-Time Fourier Transform (DTFT).
In this chapter, we limit our analysis to periodic continuous-time signals and the corresponding Fourier series
representation for them. The next chapter will deal with the Fourier transform for aperiodic signals as the period
extends to infinity.

THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS
It is advantageous in the study of LTI systems to represent signals as linear combinations of basic signals that
possess the following two properties:
1. The set of basic signals can be used to construct a broad and useful class of signals.
2. The response of an LTI system to each signal should be simple enough in structure to provide us with a
convenient representation for the response of the system to any signal constructed as a linear combination of
the basic signals.
Remark:
The importance of complex exponentials in the study of LTI systems stems from the fact that the response of an
LTI system to a complex exponential input is the same complex exponential with only a change in amplitude;
that is,
Continuous time: 𝑒𝑠𝑡
՜ 𝐻 𝑠 𝑒𝑠𝑡
; s is complex variable
Discrete time: 𝑧𝑛
՜ 𝐻 𝑧 𝑧𝑛
; z is complex variable
where the complex amplitude factor H(s) or H(z) will in general be a function of the complex variable s or z.

EXPONENTIALS
The complex exponentials are indeed eigenfunctions
of LTI systems, let us consider a continuous-time LTI
system with impulse response h(t). For an input x(t), we
can determine the output through the use of the
convolution integral, so that with 𝑥 𝑡 = 𝑒𝑠𝑡
𝑦 𝑡 = න
−∞
∞
ℎ 𝜏 𝑥 𝑡 − 𝜏 𝑑𝜏 = න
−∞
∞
ℎ 𝜏 𝑒𝑠(𝑡−𝜏)
𝑑𝜏
𝑦 𝑡 = 𝑒𝑠𝑡 න
−∞
∞
ℎ 𝜏 𝑒−𝑠𝜏
𝑑𝜏
𝑦 𝑡 = 𝐻 𝑠 𝑒𝑠𝑡
𝐻 𝑠 = ‫׬‬−∞
∞
ℎ 𝜏 𝑒−𝑠𝜏
𝑑𝜏
Hence, we have shown that complex exponentials are
eigenfunctions of LTI systems.
The constant 𝐻 𝑠 for a specific value of 𝑠 is then the
eigenvalue associated with the eigenfunction 𝑒𝑠𝑡
.
In an exactly parallel manner, we can show that
complex exponential sequences are eigenfunctions of
discrete-time LTI systems. That is, suppose that an
LTI system with impulse response h[n] has as its input
the sequence 𝑥 𝑛 = 𝑧𝑛
.
𝑦 𝑛 = ෍
𝑘=−∞
∞
ℎ 𝑘 𝑥 𝑛 − 𝑘 = ෍
𝑘=−∞
∞
ℎ 𝑘 𝑧𝑛−𝑘
𝑦 𝑛 = 𝑧𝑛 ෍
𝑘=−∞
∞
ℎ 𝑘 𝑧−𝑘
𝑦 𝑛 = 𝐻(𝑧)𝑧𝑛
𝐻(𝑧) = ෍
𝑘=−∞
∞
ℎ 𝑘 𝑧−𝑘

EXPONENTIALS
Let 𝑥(𝑡) correspond to a linear combination of three
complex exponentials; that is,
𝑥 𝑡 = 𝑎1𝑒𝑠1𝑡 + 𝑎2𝑒𝑠2𝑡 + 𝑎3𝑒𝑠3𝑡
From the eigenfunction property, the response to each
separately is
𝑎1𝑒𝑠1𝑡
՜ 𝑎1𝐻 𝑠1 𝑒𝑠1𝑡
𝑎2𝑒𝑠2𝑡
՜ 𝑎2𝐻 𝑠2 𝑒𝑠2𝑡
𝑎3𝑒𝑠3𝑡 ՜ 𝑎3𝐻 𝑠3 𝑒𝑠3𝑡
And from the superposition property the response to
the sum is the sum of the responses, so that
𝑦 𝑡 = 𝑎1𝐻(𝑠1)𝑒𝑠1𝑡 + 𝑎2𝐻(𝑠2)𝑒𝑠2𝑡 + 𝑎3𝐻(𝑠3)𝑒𝑠3𝑡
If the input to a continuous-time LTI system is
represented as a linear combination of complex
exponentials, that is, if
𝑥 𝑡 = ෍
𝑘
𝑎𝑘𝑒𝑠𝑘𝑡
Then the output will be
𝑦 𝑡 = ෍
𝑘
𝑎𝑘𝐻 𝑠𝑘 𝑒𝑠𝑘𝑡

FOURIER SERIES REPRESENTATION OF CONTINUOUS-
TIME PERIODIC SIGNALS
Linear Combinations of Harmonically Related
Complex Exponentials
• The set of harmonically related complex
exponentials are
∅𝑘 = 𝑒𝑗𝑘𝜔0𝑡
, k = 0, ±1, ±2, …
• Each of these signals has a fundamental frequency
that is a multiple of 𝜔0 =
2𝜋
𝑇0
, and therefore, each is
periodic with period
𝑇0
𝑘
(although for 𝑘 ≥ 2, the
fundamental period of ∅𝑘is a fraction of 𝑇0).
• Fourier series representation is a linear
combination of harmonically related complex
exponentials of the form
• 𝑥 𝑡 = σ𝑘=−∞
∞
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡
, 𝑤ℎ𝑒𝑟𝑒 𝜔0 = Τ
2𝜋 𝑇0
• In the CT case, all of the harmonically related
complex exponentials {∅k(t)} are distinct.
• 𝑥 𝑡 is periodic with period 𝑇0.
• The components for 𝑘 = +𝑁 and 𝑘 = − 𝑁 are
referred to as the Nth harmonic components.
• The term for 𝑘 = 0 is a constant.
• The terms for 𝑘 = +1 and 𝑘 = −1 both have
fundamental frequency equal to wo and are
collectively referred to as the fundamental
components or the first harmonic components.
• The two terms fork 𝑘 = +2 and 𝑘 = −2 are
periodic with half the period (or, equivalently, twice
the frequency) of the fundamental components and
are referred to as the second harmonic components.

Suppose that 𝑥 𝑡 is real and can be represented in the
form
𝑥 𝑡 = ෍
𝑘=−∞
∞
Then, since for a real signal 𝑥∗
(𝑡) = 𝑥(𝑡), we obtain
𝑥∗
𝑡 = ෍
𝑘=−∞
∞
𝑎𝑘
∗
𝑒−𝑗𝑘𝜔0𝑡
Replacing 𝑘 by − 𝑘 in the summation, we have
𝑥∗ 𝑡 = ෍
𝑘=−∞
∞
𝑎−𝑘
∗
𝑒𝑗𝑘𝜔0𝑡
since 𝑥∗
(𝑡) = 𝑥(𝑡), Therefore,
𝑎𝑘 = 𝑎−𝑘
∗
To derive the alternative forms of the Fourier series
𝑥 𝑡 = 𝑎0 + ෍
𝑘=1
∞
[𝑎𝑘𝑒𝑗𝑘𝜔0𝑡
+ 𝑎−𝑘
∗
𝑒−𝑗𝑘𝜔0𝑡
]
𝑥 𝑡 = 𝑎0 + ෍
𝑘=1
∞
[𝑎𝑘𝑒𝑗𝑘𝜔0𝑡
+ 𝑎𝑘𝑒−𝑗𝑘𝜔0𝑡
]
∴ 𝑥 𝑡 = 𝑎0 + ෍
𝑘=1
∞
2𝑅𝑒{𝑎𝑘𝑒𝑗𝑘𝜔0𝑡
}
If 𝑎𝑘 is expressed in polar form as 𝑎𝑘 = 𝑎𝑘 𝑒𝑗𝜃𝑘
𝑥 𝑡 = 𝑎0 + ෍
𝑘=1
∞
2𝑅𝑒{ 𝑎𝑘 𝑒𝑗𝜃𝑘𝑒𝑗𝑘𝜔0𝑡
}

𝑥 𝑡 = 𝑎0 + 2 ෍
𝑘=1
∞
𝑅𝑒{ 𝑎𝑘 𝑒𝑗(𝑘𝜔0𝑡+𝜃𝑘)}
∴ 𝑥 𝑡 = 𝑎0 + 2 ෍
𝑘=1
∞
𝑎𝑘 𝑐𝑜𝑠(𝑘𝜔0𝑡 + 𝜃𝑘)}
If we let 𝑎𝑘= 𝑏𝑘 + 𝑗𝑐𝑘, we get
∴ 𝑥 𝑡 = 𝑎0 + 2 ෍
𝑘=1
∞
[𝑏𝑘𝑐𝑜𝑠 𝑘𝜔0𝑡 + 𝜃𝑘 − 𝑐𝑘𝑠𝑖𝑛 𝑘𝜔0𝑡 + 𝜃𝑘 ]

Determination of the Fourier Series
Representation of a Continuous-time Periodic
Signal
𝑥 𝑡 = ෍
𝑘=−∞
∞
Multiply both sides by 𝑒−𝑗𝑛𝜔0𝑡 we get,
𝑒−𝑗𝑛𝜔0𝑡 𝑥 𝑡 = σ𝑘=−∞
∞
𝑎𝑘𝑒−𝑗𝑛𝜔0𝑡𝑒𝑗𝑘𝜔0𝑡
න
𝑇0
𝑒−𝑗𝑛𝜔0𝑡
𝑥 𝑡 𝑑𝑡 = ෍
𝑘=−∞
∞
𝑎𝑘[න
𝑇0
𝑒𝑗(𝑛−𝑘)𝜔0𝑡
𝑑𝑡]
The evaluation of the bracketed integral is
straightforward. Rewriting this integral using
Euler's formula, we obtain
න
𝑇0
𝑒𝑗(𝑛−𝑘)𝜔0𝑡𝑑𝑡 = න
𝑇0
𝑐𝑜𝑠 𝑘 − 𝑛 𝜔0𝑡 𝑑𝑡
+𝑗 ‫׬‬𝑇0
𝑠𝑖𝑛 𝑘 − 𝑛 𝜔0𝑡 𝑑𝑡
න
𝑇0
𝑒𝑗(𝑛−𝑘)𝜔0𝑡𝑑𝑡 = ቊ
𝑇0, 𝑘 = 𝑛
0, 𝑘 ≠ 𝑛
∴ 𝑎𝑛=
1
𝑇0
න
𝑇0
𝑥 𝑡 𝑒−𝑗𝑛𝜔0𝑡
𝑑𝑡
which provides the equation for determining the
coefficients 𝑎𝑘.

Remark:
• Two signals are orthogonal if their inner product is zero.
• The inner product between two harmonically related complex exponentials ∅𝑘 𝑡 and ∅𝑛 𝑡 is defined as :
න
𝑇0
∅𝑘 𝑡 ∅𝑛
∗
𝑡 𝑑𝑡 = න
𝑇0
𝑒𝑗𝑘𝜔0𝑡
𝑒−𝑗𝑛𝜔0𝑡
𝑑𝑡
= ቊ
𝑇0, 𝑘 = 𝑛
0, 𝑘 ≠ 𝑛

The Fourier series of a periodic continuous-time signal:
𝑥 𝑡 = σ𝑘=−∞
∞
= 𝑎0 + 2 σ𝑘=1
∞
𝑎𝑘 𝑐𝑜𝑠(𝑘𝜔0𝑡 + 𝜃𝑘)} …(*)
= 𝑎0 + 2 σ𝑘=1
∞
[𝑏𝑘𝑐𝑜𝑠 𝑘𝜔0𝑡 + 𝜃𝑘 − 𝑐𝑘𝑠𝑖𝑛 𝑘𝜔0𝑡 + 𝜃𝑘 ] …(**)
𝑎𝑘 =
1
𝑇0
‫׬‬𝑇0
𝑥 𝑡 𝑒−𝑗𝑘𝜔0𝑡𝑑𝑡 = 𝑏𝑘 + 𝑗𝑐𝑘 …(***)
(The set of coefficients {𝑎𝑘} are often called the Fourier series coefficients or the spectral coefficients of 𝑥 𝑡 .)
It follows that:
𝑏𝑘 =
1
𝑇0
‫׬‬𝑇0
𝑥 𝑡 𝑐𝑜𝑠 𝑘𝜔0𝑡 𝑑𝑡 ; 𝑐𝑘 =
−1
𝑇0
‫׬‬𝑇0
𝑥 𝑡 𝑠𝑖𝑛 𝑘𝜔0𝑡 𝑑𝑡 ;
𝑎0 =
1
𝑇0
න
𝑇0
𝑥 𝑡 𝑑𝑡 (𝑎0 is simply the average value of 𝑥 𝑡 over one period.)

Example 3.3 :
Consider the signal 𝑥 𝑡 = sin 𝜔0𝑡 .
whose fundamental frequency is 𝜔0 . One approach to determining the Fourier series coefficients for this signal
is to apply the analysis equation (***). For this simple case, however, it is easier to expand the sinusoidal signal
as a linear combination of complex exponentials and identify the Fourier series coefficients by inspection.
Specifically, we can express sin 𝜔0𝑡 as
sin 𝜔0𝑡 =
(𝑒𝑗𝜔0𝑡
− 𝑒−𝑗𝜔0𝑡
)
2𝑗
Comparing the right-hand sides of this equation and synthesis equation (*) , we obtain
∴ 𝑎1 =
1
2𝑗
; 𝑎−1 =
−1
2𝑗
𝑎𝑘 = 0, 𝑘 ≠ +1 𝑜𝑟 − 1

Example 3.4 :
Let
𝑥 𝑡 = 1 + sin 𝜔0𝑡 + 2 cos 𝜔0𝑡 + cos 2𝜔0𝑡 +
𝜋
4
which has fundamental frequency 𝜔0. As with Example 3.3, we can again expand 𝑥(𝑡) directly in terms of
complex exponentials, so that
𝑥 𝑡 = 1 +
1
2𝑗
(𝑒𝑗𝜔0𝑡
− 𝑒−𝑗𝜔0𝑡
) + (𝑒𝑗𝜔0𝑡
+ 𝑒−𝑗𝜔0𝑡
) +
1
2
(𝑒
𝑗 2𝜔0𝑡+
𝜋
4 + 𝑒
−𝑗 2𝜔0𝑡+
𝜋
4 )
Collecting terms, we obtain
𝑥 𝑡 = 1 + (1 +
1
2𝑗
)𝑒𝑗𝜔0𝑡
+(1 −
1
2𝑗
)𝑒−𝑗𝜔0𝑡
+
1
2
𝑒𝑗
𝜋
4𝑒𝑗2𝜔0𝑡
+
1
2
𝑒−𝑗
𝜋
4𝑒−𝑗2𝜔0𝑡

Thus, the Fourier series coefficients for this example
are

• Examples : 3.3,3.4, and 3.5, Oppenheim, Willsky, and Nawab, 2nd edition.
• Suggested problems:
chapter 3, basic problems: 3.3, 3.4, and 3.8.

CONVERGENCE OF THE FOURIER SERIES
• In fact, Fourier maintained that any periodic signal could be represented by a Fourier series. Although this is
not quite true, it is true that Fourier series can be used to represent an extremely large class of periodic
signals, including the square wave and all other periodic signals with which we will be concerned in this book
and which are of interest in practice.
• Let us examine the problem of approximating a given periodic signal 𝑥(𝑡) by a linear combination of a finite
number of harmonically related complex exponentials-that is, by a finite series of the form
𝑥𝑁 𝑡 = σ𝑘=−𝑁
+𝑁
• Let 𝑒𝑁 𝑡 denote the approximation error; that is
𝑒𝑁 𝑡 = 𝑥 𝑛 − 𝑥𝑁 𝑡 = 𝑥 𝑛 − σ𝑘=−𝑁
+𝑁
• In order to determine how good any particular approximation is, we need to specify a quantitative measure of
the size of the approximation error. The criterion that we will use is the energy in the error over one period:
𝐸𝑁 = න
𝑇
𝑒𝑁(𝑡) 2
𝑑𝑡

Remark:
As shown in Problem 3.66 (extension problem), the particular choice for the coefficients in 𝑥𝑁 𝑡 =
σ𝑘=−𝑁
+𝑁
that minimize the energy in the error is
𝑎𝑘 =
1
𝑇
න
𝑇
𝑥 𝑡 𝑒−𝑗𝑘𝜔0𝑡
𝑑𝑡
Thus, if 𝑥(𝑡) has a Fourier series representation, the best approximation using only a finite number of
harmonically related complex exponentials is obtained by truncating the Fourier series to the desired number of
terms.
As 𝑁 increases, new terms are added and 𝐸𝑁 decreases. If, in fact, 𝑥(𝑡) has a Fourier series representation, then
the limit of 𝐸𝑁 as N՜ ∞ is zero

• Let us tum now to the question of when a periodic signal 𝑥(𝑡) does in fact have a Fourier series
representation.
• Every continuous periodic signal has a Fourier series representation for which the energy 𝑬𝑵 is zero in
the approximation error approaches 0 as N goes to ∞. This is also true for many discontinuous signals.
• Since we will find it very useful to include discontinuous signals such as square waves in our discussions, it
is worthwhile to investigate the issue of convergence in a bit more detail.
• One class of periodic signals that are representable through the Fourier series is those signals which have
finite energy over a single period, i.e., signals 𝑥 𝑡 for which is square integrable.
න
𝑇0
𝑥(𝑡) 2
𝑑𝑡 < ∞
If 𝑥 𝑡 is square integrable ‫׬‬𝑇0
𝑥(𝑡) 2𝑑𝑡 < ∞, 𝐭hen ‫׬‬𝑇
𝑒𝑁(𝑡) 2𝑑𝑡 ՜ 0 as 𝑁 ՜ ∞.
‫׬‬𝑇
𝑒𝑁(𝑡) 2𝑑𝑡 = 0 does not imply that the signal 𝑥(𝑡) and its Fourier series representation 𝑥 𝑡 =
σ𝑘=−∞
∞
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡 are equal at every value of t. What it does say is that there is no energy in their difference.

An alternative set of conditions, developed by P. L. Dirichlet and also satisfied by essentially all of the signals
with which we will be concerned, guarantees that 𝑥(𝑡) equals its Fourier series representation, except at isolated
values of 𝑡 for which 𝑥(𝑡) is discontinuous. At these values, the infinite series σ𝑘=−∞
∞
converges to the
average of the values on either side of the discontinuity.
The Dirichlet conditions are as follows:
Condition 1. Over any period, 𝑥(𝑡) must be absolutely integrable; that is, ‫׬‬𝑇
𝑥(𝑡)𝑑𝑡 < ∞
Condition 2. In any finite interval of time, 𝑥(𝑡) is of bounded variation; that is, there are no more than a finite
number of maxima and minima during any single period of the signal.
Condition 3. In any finite interval of time, there are only a finite number of discontinuities. Furthermore, each
of these discontinuities is finite.

For a periodic signal that has no discontinuities, the Fourier series representation converges and equals the
original signal at every value of 𝑡. For a periodic signal with a finite number of discontinuities in each
period, the Fourier series representation equals the signal every where except at the isolated points of
discontinuity, at which the series converges to the average value of the signal on either side of the discontinuity.
In this case the difference between the original signal and its Fourier series representation contains no energy,
and consequently, the two signals can be thought of as being the same for all practical purposes. Specifically,
since the signals differ only at isolated points, the integrals of both signals over any interval are identical. For
this reason, the two signals behave identically under convolution and consequently are identical from the
standpoint of the analysis of LTI systems.

In 1898, an American physicist, Albert Michelson,
constructed a harmonic analyzer, a device that, for any
periodic signal 𝑥(𝑡), would compute the truncated
Fourier series approximation of σ𝑘=−𝑁
+𝑁
for
values of 𝑁 up to 80. The interesting effect that
Michelson observed is that the behavior of the partial
sum in the vicinity of the discontinuity exhibits
ripples and that the peak amplitude of these ripples
does not seem to decrease with increasing 𝑁. Gibbs
showed that these are in fact the case. Specifically, for
a discontinuity of unity height, the partial sum
exhibits a maximum value of 1.09 (i.e., an overshoot
of 9% of the height of the discontinuity), no matter
how large N becomes
An illustration of the Gibbs phenomenon

TIME PERIODIC SIGNALS (Signals and Systems, S.Palani, page 435)

PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
• Fourier series representations possess a number of
important properties that are useful for developing
conceptual insights into such representations,
and they can also help to reduce the complexity of
the evaluation of the Fourier series of many signals.
• In Chapter 4, in which we develop the Fourier
transform, we will see that most of these properties
can be deduced from corresponding properties of
the continuous-time Fourier transform.
• Suppose that 𝑥(𝑡) is a periodic signal with period T
and fundamental frequency 𝑤0 =
2𝜋
𝑇
. Then if the
Fourier series coefficients of 𝑥(𝑡) are denoted by
𝑎𝑘 we will use the notation
to signify the pairing of a periodic signal with its
Fourier series coefficients.

1. Linearity
Let 𝑥(𝑡) and 𝑦(𝑡) denote two periodic signals with
period 𝑇 and which have Fourier series coefficients
denoted by 𝑎𝑘 and 𝑏𝑘 , respectively. That is,
Since 𝑥(𝑡) and 𝑦(𝑡) have the same period 𝑇, it easily
follows that any linear combination of the two signals
will also be periodic with period T.
Furthermore, the Fourier series coefficients 𝑐𝑘 , of the
linear combination of 𝑥(𝑡) and 𝑦(𝑡) , 𝑧(𝑡) =
𝐴𝑥(𝑡) + 𝐵𝑦(𝑡), are given by the same linear
combination of the Fourier series coefficients for 𝑥(𝑡)
and 𝑦(𝑡). That is,

2. Time Shifting
• When a time shift is applied to a periodic signal
𝑥(𝑡), the period 𝑻 of the signal is preserved.
• The Fourier series coefficients 𝑏𝑘 of the resulting
signal 𝑦(𝑡) = 𝑥(𝑡 − 𝑡0) may be expressed
as
Letting 𝜏 = 𝑡 − 𝑡0in the integral, and noting that the
new variable 𝜏 will also range over an interval of
duration 𝑇, we obtain
where 𝑎𝑘 is the 𝑘𝑡ℎ Fourier series coefficient of 𝑥(𝑡).
That is, if
then,
when a periodic signal is shifted in time, the
magnitudes of its Fourier series coefficients remain
unaltered. That is, 𝒃𝒌 = 𝒂𝒌 ·

3. Time Reversal
• The period 𝑇 of a periodic signal 𝑥(𝑡) also remains
unchanged when the signal undergoes time
reversal. To determine the Fourier series
coefficients of 𝑦(𝑡) = 𝑥(− 𝑡), let us consider the
effect of time reversal on the synthesis equation.
• Making the substitution 𝑘 = − 𝑚 , we obtain
• The Fourier series coefficients 𝑏𝑘 of 𝑦(𝑡)
= 𝑥(− 𝑡) are
That is, if
then
• Time reversal applied to a continuous-time signal
results in a time reversal of the corresponding
sequence of Fourier series coefficients.

If 𝑥(𝑡) is even-that is, if 𝒙(−𝒕) = 𝒙(𝒕) −then its
Fourier series coefficients are also even-i.e., 𝒂−𝒌
= 𝒂𝒌 Similarly, if 𝑥(𝑡) is odd, so that 𝒙( −𝒕)
= − 𝒙(𝒕), then so are its Fourier series coefficients-
i.e., 𝒂−𝒌 = −𝒂𝒌.

4. Time Scaling
• Time scaling is an operation that in general changes
the period of the underlying signal.
• If 𝑥(𝑡) is periodic with period T and fundamental
frequency 𝑤0 =
2𝜋
𝑇
, then 𝑥(𝛼𝑡), where a is a
positive real number, is periodic with period
𝑇
𝛼
and
fundamental frequency 𝛼𝜔0 .
• That is, if 𝑥(𝑡) has the Fourier series representation,
then the Fourier series representation of 𝑥(𝛼𝑡) is
• The Fourier coefficients have not changed, the
Fourier series representation has changed because
of the change in the fundamental frequency.

5. Multiplication
Suppose that 𝑥(𝑡) and 𝑦(𝑡) are both periodic with
period 𝑇 and that
Since the product 𝑥(𝑡)𝑦(𝑡) is also periodic with
period 𝑇, we can expand it in a Fourier series with
Fourier series coefficients ℎ𝑘, expressed in terms of
those for 𝑥(𝑡) and 𝑦(𝑡). The result is
• The sum on the right-hand side of this sum ℎ𝑘 may
be interpreted as the discrete-time convolution of
the sequence representing the Fourier coefficients
of 𝑥(𝑡) and the sequence representing the Fourier
coefficients of 𝑦(𝑡).
= 𝑎𝑘∗ 𝑏𝑘

6. Conjugation and Conjugate Symmetry
• Taking the complex conjugate of a periodic signal
𝑥(𝑡) has the effect of complex conjugation and
time reversal on the corresponding Fourier series
coefficients. That is, if
then
• This property is easily proved by applying complex
conjugation to both sides of the synthesis equation
and replacing the summation variable k by its
negative.
• For 𝑥(𝑡) real-that is, when 𝑥 𝑡 = 𝑥∗ 𝑡 , the
Fourier series coefficients will be conjugate
symmetric, i.e.,
• This in turn implies various symmetry properties
(listed in Table 3.1) for the magnitudes, phases, real
parts, and imaginary parts of the Fourier series
coefficients of real signals.
• If 𝒙(𝒕) is real, then 𝒂𝟎 is real and 𝒂𝒌 = 𝒂−𝒌 ·

7. Parseval's Relation for Continuous-Time
Periodic Signals
• Parseval's relation for continuous-time periodic
signals is
where the 𝑎𝑘 are the Fourier series coefficients of
𝑥(𝑡) and Tis the period of the signal
• Note that the left-hand side of the Parseval's
relation is the average power (i.e., energy per unit
time) in one period of the periodic signal 𝑥(𝑡).
Also,
• So that 𝑎𝑘
2 is the average power in the kth
harmonic component of 𝑥(𝑡). Thus, what Parseval's
relation states is that the total average power in a
periodic signal equals the sum of the average
powers in all of its harmonic components.

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