566
R. B. Pachori and V. Gupta
the analysis of a signal. Therefore, the Fourier transform is a commonly used tool
for spectral representation of a time-domain signal. The main motivation behind
the uses of different types of transformations in signal processing techniques is due
to the fact that transforms can highlight certain characteristics present in signal in
different domains.
According to Fourier, a continuous periodic signal x(t) can be formed by
combining a number of scaled and phase-shifted sinusoidal components. The
frequencies of these components are in multiple of the fundamental frequency (ω 0 )
for the signal x(t). Hence, the synthesis equation for a general periodic signal x(t)
can be written as [12]:
x(t) =
∞
k=0
g k cos
2πkf 0 t + k
(12.24)
where g k and k are sets of constants and f 0 =
ω 0
2π . Suppose p k = g k cos ( k ) and
q k = − g k sin ( k ), then Eq. (12.24) with the help of trigonometric expansion can
be written as [12]:
x(t) =
∞
k=0
[p k cos (2πf k t) + q k sin (2πf k t)]
(12.25)
Equation (12.25) can also be written as [12]:
x(t) =
∞
k=−∞
A k [cos (ω k t) + j. sin (ω k t)]
(12.26)
where j =
√
−1 , A k =
p k ±jq k
2
, and it is a complex number for k > 0 and k < 0,
respectively. Equation (12.26) with the help of Euler’s relation can be written as
[12]
x(t) =
∞
k=−∞
A k e
+jkω 0 t
(12.27)
Here, the magnitude of coefficient, |A k | = g k =
p 2
k + q 2
k
and phase
A k = k = tan −1 (q k /p k ). Equation (12.27) is known as Fourier synthesis equation
for a periodic continuous signal x(t).
Conversely, the Fourier analysis equation for a periodic continuous signal x(t)
with time period T 0 can be written as [12]:
A k =
1
T 0
x(t)e
−jkω 0 t dt
(12.28)