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The representation and analysis of LTI systems through the convolution sum
is based on representing signals as linear combinations of time-shifted
impulses.
An alternative representation for signals and LTI systems is using
sinusoids/complex exponentials. The resulting representations are known as
the continuous-time and discrete-time Fourier series and transform. These
representations can be used to construct broad and useful classes of signals.
Representation of continuous-time and discrete-time periodic signals using
sinusoids/complex exponentials is referred to as the Fourier series.
Representation of continuous-time and discrete-time aperiodic finite energy
signals using sinusoids/complex exponentials is referred to as the Fourier
transform.
History :
The modern history of the subject begins in 1748 with L.
Euler , who examined the motion of a vibrating string. For
any fixed instant of time (t), the normal modes are
harmonically related sinusoidal functions of x. Euler
noted that if the configuration of a vibrating string at
some point in time is a linear combination of these normal
modes, so is the configuration at any subsequent time.
Euler could calculate the coefficients for the linear
combination at the later time in a very straightforward
manner from the coefficients at the earlier time. The same
type of calculation will be performed in deriving one of
the properties of trigonometric sums that make them so
useful for the analysis of LTI systems.
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 as a linear
combination of periodic complex exponentials or
sinusoids, with coefficients that are related in a
straightforward way to those of the input.
In 1753, D. Bernoulli argued on physical grounds that all
physical motions of a string could be represented by
linear combinations of normal modes (without mathematical analysis). His ideas were not widely accepted. 1759 J. L.
Lagrange strongly criticized the use of trigonometric series in the examination of vibrating strings.
In this hostile and sceptical environment that Jean Baptiste Joseph Fourier presented his ideas half a century later. In
1807, Fourier had found series of harmonically related sinusoids to be useful in representing the temperature distribution
through a body. In addition, he claimed that "any" periodic signal could be represented by such a series.
There are four distinct Fourier representations, each applicable to a different class of signals. These four classes are defined
by the periodicity properties of a signal and whether it is continuous-time or discrete-time.
A periodic signal 𝑥(𝑡) with period 𝑇 𝑜
has the property
𝑜
) for all 𝑡
The smallest value of 𝑇 𝑜
that satisfies this periodicity condition is the fundamental period of 𝑥(𝑡). this equation implies
that 𝑥(𝑡) starts at −∞ and continues to +∞. The area under a periodic signal 𝑥(𝑡) over any interval of duration 𝑇 𝑜
is the
same.
𝑎+𝑇 𝑜
𝑎
𝑏+𝑇 𝑜
𝑏
Sinusoids : The frequency of a sinusoid 𝑐𝑜𝑠 2 𝜋𝑓
𝑜
𝑡 or 𝑠𝑖𝑛 2 𝜋𝑓
𝑜
𝑡 is 𝑓
𝑜
, and the period is 𝑇
𝑜
𝑜
. A sinusoid of frequency
𝑜
is said to be the 𝒏
𝒕𝒉
harmonic of the sinusoid of frequency 𝑓
𝑜
Fourier Representation of Continuous-Time Periodic Signals
Trigonometric Fourier Spectra
The compact form of Trigonometric Fourier Series indicates that a periodic signal 𝑥(𝑡) can be expressed as a sum of
sinusoids of frequencies 0 (DC), 𝜔 𝑜
𝑜
𝑜
, … , whose amplitudes are 𝐶
𝑜
1
2
𝑛
, …, and whose phases
are 0, 𝜃 1
2
3
𝑛
, …, respectively. i.e.,
𝑜
𝑛
cos
𝑜
𝑛
∞
𝑛= 1
𝑜
1
cos
𝑜
1
2
cos
𝑜
2
𝑛
cos
𝑜
𝑛
Amplitude Spectrum: Plot of amplitude 𝐶 𝑛
versus 𝑛
Phase Spectrum: Plot of phase 𝜃 𝑛
versus 𝑛
The amplitude and phase spectra are scaled plots of 𝐶 𝑛
versus 𝜔 and 𝜃
𝑛
versus 𝜔. (since 𝑛 𝛼 𝑛𝜔
𝑜
Amplitude Spectrum + Phase Spectrum = frequency spectra of x(t)
Frequency spectra are an alternative way of describing a periodic signal 𝑥(𝑡), and are in every way equivalent to the plot
of 𝑥(𝑡) as a function of 𝑡.
The amplitude and phase spectra for 𝑥(𝑡) tell us at a glance the frequency composition of x(t) , that is, the amplitudes and
phases of various sinusoidal components of 𝑥(𝑡). Knowing the frequency spectra, we can reconstruct 𝑥(𝑡). The frequency
spectra provide an alternative description − the frequency-domain description of x(t).
A signal, has a dual identity: the time-domain identity [𝑥(𝑡)] and the frequency-domain identity [ Fourier spectra ]. The
two identities taken together provide a better understanding of a signal.
Exponential Fourier Series
The exponential Fourier series for a periodic signal 𝑥(𝑡) can be expressed as
𝑛
𝑗𝑛𝜔
𝑜
𝑡
∞
𝑛=−∞
where
𝑛
𝑜
−𝑗𝑛𝜔
𝑜
𝑡
<𝑇
𝑜
Exponential Fourier Spectra
In exponential Fourier spectra, we plot coefficients 𝐷 𝑛
as a function of 𝜔. Since 𝐷
𝑛
is complex in general, we need both
parts of one of two sets of plots: the real and the imaginary parts of 𝐷 𝑛
, or the magnitude and the angle of 𝐷
𝑛
. The
exponential Fourier Series indicates that a periodic signal 𝑥(𝑡) can be expressed as a sum of complex exponentials of
frequencies 0 (DC), ±𝜔 𝑜
𝑜
𝑜
, … , whose magnitudes are |𝐷
𝑜
± 1
± 2
±𝑛
|, …, and whose
phases are 0 , ∠𝐷
± 1
± 2
±𝑛
, …, respectively. i.e.,
𝑛
𝑗∠𝐷
𝑛
𝑒
𝑗𝑛𝜔
𝑜
𝑡
∞
𝑛=−∞
− 2
𝑗∠𝐷 − 2
−𝑗 2 𝜔 𝑜
𝑡
− 1
𝑗∠𝐷 − 1
−𝑗 1 𝜔 𝑜
𝑡
0
1
𝑗∠𝐷 1
𝑗 1 𝜔 𝑜
𝑡
2
𝑗∠𝐷 2
𝑗 2 𝜔 𝑜
𝑡
Note : For real 𝑥(𝑡), the magnitude spectrum (|𝐷 𝑛
| versus 𝜔) is an even function of 𝜔 and the phase spectrum (∠𝐷
𝑛
versus
𝜔) is an odd function of 𝜔.
i,e.,
−𝑛
𝑛
and ∠𝐷
−𝑛
𝑛
The Effect of Symmetry
The Fourier series of any even periodic signal 𝑥(𝑡) consists of cosine terms only and the series for any odd periodic signal
𝑥(𝑡) consists of sine terms only. Knowing the signal over a half-period and knowing the kind of symmetry (even or odd),
we can determine the signal waveform over a complete period. Hence, the Fourier coefficients in these cases can be
computed by integrating over only half the period rather than a complete period.
𝑜
𝑜
𝑇
𝑜
/ 2
−𝑇
𝑜
/ 2
𝑛
𝑜
cos
𝑜
𝑇
𝑜
/ 2
−𝑇
𝑜
/ 2
𝑛
𝑜
sin (𝑛𝜔
𝑜
𝑇 𝑜
/ 2
−𝑇
𝑜
/ 2
If 𝑥
is an even function of 𝑡, then 𝑥
𝑜
𝑡 is also an even function and 𝑥
𝑜
𝑡 is an odd function of 𝑡.
Therefore,
𝑜
2
𝑇 𝑜
𝑇
𝑜
/ 2
0
𝑛
4
𝑇 𝑜
cos
𝑜
𝑇
𝑜
/ 2
0
𝑛
Example:
If 𝑥
is an odd function of 𝑡, then 𝑥(𝑡)𝑐𝑜𝑠𝑛𝜔
𝑜
𝑡 is an odd function and 𝑥
𝑜
𝑡 is an even function of 𝑡. Therefore,
𝑛
𝑛
4
𝑇
𝑜
sin
𝑜
𝑇
𝑜
/ 2
0
Example:
If a periodic signal 𝑥(𝑡), with period 𝑇 𝑜
, shifted by half the period remains unchanged except for a sign—that is, if
𝑜
then the signal is said to have a half-wave symmetry. For a signal with a half-wave symmetry, all the even-numbered
harmonics vanish.
Example:
Symmetry Condition Trigonometric Fourier Series Coefficients
Even
𝑜
𝑜
𝑇
𝑜
/ 2
0
𝑛
𝑜
cos
𝑜
𝑇
𝑜
/ 2
0
𝑛
Odd
𝑛
𝑛
𝑜
sin
𝑜
𝑇
𝑜
/ 2
0
Half-wave+ Even
𝑜
and
𝑛
4
𝑇
𝑜
𝑥(𝑡) cos(𝑛𝜔
𝑜
𝑇
𝑜
/ 2
0
𝑛
Half-wave+ Odd
𝑜
and
𝑛
𝑛
4
𝑇 𝑜
sin
𝑜
𝑇
𝑜
/ 2
0
PROPERTIES OF CONTINUOUS-TIME EXPONENTIAL FOURIER SERIES
Property Periodic Signal
Fourier Series
Coefficients
𝑥(𝑡)
𝑦
( 𝑡
)
}
Periodic with period 𝑇
𝑜
and fundamental frequency 𝜔
𝑜
= 2 𝜋/𝑇
𝑜
𝐷
𝑛
𝐸
𝑛
Linearity 𝐴𝑥
( 𝑡
)
𝑛
𝑛
Time Shifting 𝑥
( 𝑡 − 𝑡
𝑜
) 𝑒
−𝑗𝑘𝜔
𝑜
𝑡
𝑜
𝐷
𝑛
Frequency Shifting
𝑒
𝑗𝑛
𝑜
𝜔
𝑜
𝑡
𝑥
( 𝑡
) 𝐷
𝑛−𝑛
𝑜
Conjugation 𝑥
∗
( 𝑡
) 𝐷
−𝑛
∗
Time Reversal 𝑥
( −𝑡
) 𝐷
−𝑛
Time Scaling 𝑥
( 𝛼𝑡
) , 𝛼 > 0 (periodic with period 𝑇
𝑜
/𝛼) 𝐷
𝑛
Periodic
Convolution
∫ 𝑥
( 𝜏
) 𝑦
( 𝑡 − 𝜏
) 𝑑𝜏
<𝑇
𝑜
𝑇
𝑜
𝐷
𝑛
𝐸
𝑛
Multiplication
𝑥(𝑡)𝑦(𝑡) ∑ 𝐷
𝑘
𝐸
𝑛−𝑘
∞
𝑘=−∞
Differentiation
𝑑
𝑑𝑡
𝑥(𝑡)
𝑗𝑛𝜔
𝑜
𝐷
𝑛
Integration
∫ 𝑥
( 𝜏
) 𝑑𝜏
𝑡
∞
(
1
𝑗𝑛𝜔
𝑜
) 𝐷
𝑛
Conjugate
Symmetry for real
signals
𝑥(𝑡) real 𝐷
𝑛
= 𝐷
−𝑛
∗
Real and Even
Signals
𝑥
( 𝑡
) real and even 𝐷
𝑛
real and even
Real and Odd
Signals
𝑥
( 𝑡
) real and odd
𝐷
𝑛
purely
imaginary and odd
Even-odd
decomposition of
real signals
𝑥
𝑒
( 𝑡
) [𝑥
( 𝑡
) real]
𝑥
𝑜
( 𝑡
) [𝑥
( 𝑡
) real]
𝑅𝑒{𝐷
𝑛
}
𝑗𝐼𝑚{𝐷
𝑛
}
Parseval’s relation for Periodic Signals
1
𝑇
∫
| 𝑥
( 𝑡
)|
2
𝑑𝑡
<𝑇
𝑜
= ∑
| 𝐷
𝑛
|
2
+∞
𝑛=−∞
Parseval’s Theorem (for continuous-time power signals)
The trigonometric Fourier series of a periodic signal 𝑥(𝑡) is given by
𝑜
𝑛
cos
𝑜
𝑛
∞
𝑛= 1
(or) 𝑥(𝑡) = 𝐶
𝑜
1
cos(𝜔
𝑜
1
2
cos( 2 𝜔
𝑜
2
3
cos( 3 𝜔
𝑜
3
𝑛
cos(𝑛𝜔
𝑜
𝑛
Every term on the right-hand side of this equation is a power signal. The power of 𝑥
is equal to the sum of the powers
of all the sinusoidal components on the right-hand side.
𝑥
𝑜
2
𝑛
2
∞
𝑛= 1
This result is one form of Parseval’s theorem , as applied to power signals. It states that the power of a periodic signal is
equal to the sum of the powers of its Fourier components.
The power of a periodic signal 𝑥(𝑡) can be expressed as a sum of the powers of its exponential components.
𝑥
𝑛
2
∞
𝑛=−∞
For a real 𝑥(𝑡) ,
−𝑛
𝑛
. Hence,
𝑥
𝑜
2
𝑛
2
∞
𝑛= 1
Fourier Representation of Discrete-Time Periodic Signals
A discrete-time signal 𝑥[𝑛] is periodic with period 𝑁 if
𝑥[𝑛 + 𝑁] = 𝑥[𝑛] for all 𝑛
The fundamental period is the smallest positive integer 𝑁 for which the equation (1) holds. Ω
o
2 𝜋
𝑁
is the
fundamental frequency.
For example, the complex exponential 𝑒
𝑗( 2 𝜋 𝑁
⁄ )𝑛
is periodic with period 𝑁. Furthermore, the set of all discrete-
time complex exponential signals that are periodic with period 𝑁 is given by
𝑘
[
]
𝑗𝑘Ω
o
𝑛
𝑗𝑘( 2 𝜋 ⁄𝑁 )𝑛
All these signals have fundamental frequencies that are multiples of 2 𝜋 ⁄𝑁 and thus are harmonically related.
There are only 𝑁 distinct signals in the set given by equation (2). This is a consequence of the fact that discrete-
time complex exponentials which differ in frequency by a multiple of 2 𝜋 are identical. Specifically, 𝜙
𝑜
[𝑛] =
𝑁
[𝑛], 𝜙
1
[𝑛] = 𝜙
𝑁+ 1
[𝑛], … and in general,
𝑘
[𝑛] = 𝜙
𝑘+𝑟𝑁
[𝑛].
That is, when 𝑘 is changed by any integer multiple of 𝑁 , the identical sequence is generated, which is different
from the situation in continuous-time in which the signals 𝜙
1
are all different from one another.
Any general discrete-time periodic signal with period 𝑁 can be represented as a linear combination of the
sequences 𝜙
𝑘
[
]
𝑗𝑘Ω o
𝑛
as
[
]
𝑘
𝑗𝑘Ω
o
𝑛
𝑘
𝑘
𝑗𝑘( 2 𝜋 ⁄𝑁 )𝑛
𝑘
Since the sequence 𝜙
𝑘
[
]
𝑗𝑘Ω o
𝑛
are distinct only over a range of 𝑁 successive values of 𝑘, the above
summation need only 𝑁 terms over this range.
[
]
𝑘
𝑗𝑘Ω
o
𝑛
𝑘=<𝑁>
𝑘
𝑗𝑘( 2 𝜋 ⁄𝑁 )𝑛
𝑘=<𝑁>
where
𝑘
[
]
−𝑗𝑘Ω o
𝑛
𝑛=<𝑁>
∑ 𝑥[𝑛]𝑒
−𝑗𝑘
( 2 𝜋 𝑁
⁄ ) 𝑛
𝑛=<𝑁>
Note: Since 𝜙
𝑘
[𝑛] is periodic with period 𝑁 (𝜙
𝑘
[𝑛] = 𝜙
𝑘+𝑟𝑁
[𝑛]), we can conclude that
𝑘
𝑘+𝑟𝑁
i.e., the DTFS coefficients 𝑎
𝑘
of a periodic signal 𝑥[𝑛] with period 𝑁 repeat periodically with period
