Analysis of Exponential Fourier Series
Fourier Series at a Glance
A continuous time signal x(t) is said to be periodic if there is a positive non-zero value of T for which
As we know any periodic signal can be classified into harmonically related sinusoids or complex exponential, provided it satisfies the Dirichlet’s Conditions
. This decomposed representation is called FOURIER SERIES
Two type of Fourier Series
representation are there. Both are equivalent to each other.
- Exponential Fourier Series
- Trigonometric Fourier Series
Both representations give the same result. Depending upon the type of signal, we choose any of the representation according to our convenience.
Exponential Fourier Series
A periodic signal is analyzed in terms of Exponential Fourier Series
in the following three stages:
- Representation of Periodic Signal.
- Amplitude and Phase Spectra of a Periodic Signal.
- Power Content of a Periodic Signal.
Representation of Periodic Signal
A periodic signal in Fourier Series may be represented in two different time domains:
- Continuous Time Domain.
- Discrete Time Domain.
Continuous Time Domain
The complex Exponential Fourier Series
representation of a periodic signal x(t) with fundamental period To
is given by
Where, C is known as the Complex Fourier Coefficient
and is given by,
, denotes the integral over any one period and, 0 to T0
/2 to T0
/2 are the limits commonly used for the integration.
The equation (3) can be derived be multiplying both sides of equation (2) by e(-jlω0t)
and integrate over a time period both sides.
On interchanging the order of summation and integration on R.H.S., we get
When, k≠l, the right hand side of (5) evaluated at the lower and upper limit yields zero. On the other hand, if k=l, we have
Consequently equation (4) reduces to
which indicates average value of x(t) over a period.
When x (t) is real,
Where, * indicates conjugate
Discrete Time Domain
Fourier representation in discrete is very much similar to Fourier representation of periodic signal of continuous time domain.
The discrete Fourier series representation of a periodic sequence x[n] with fundamental period No
is given by
, are the Fourier coefficients and are given by
This can be derived in the same way as we derived it in continuous time domain.
Amplitude and Phase Spectra of a Periodic Signal
We can express Complex Fourier Coefficient, Ck
A plot of |Ck
| versus the angular frequency w is called the amplitude spectrum of the periodic signal x(t), and a plot of Фk
, versus w is called the phase spectrum of x(t). Since the index k assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies kω0
, they are therefore referred to as discrete frequency spectra or line spectra.
For a real periodic signal x (t) we have C-k
Hence, the amplitude spectrum is an even function of ω, and the phase spectrum is an odd function of 0
for a real periodic signal.
Power Content of a Periodic Signal
Average Power Content of a Periodic Signal
is given by
If x (t) is represented by the complex exponential Fourier Series, then
This equation is known as Parseval’s identity or Parseval’s Theorem.