Signals and Systems
1
Unit-2
Analysis of Continuous Time
Signals
2
Contents
Sl No. Content
1
Introduction to Fourier series
2
Representation of Continuous time Periodic signal
3
Fourier series: Trigonometric representation
4
Fourier series: Cosine representation
5
Symmetry conditions
6
Properties of Continuous time Fourier series
7
Practice problems on Fourier series
8
Gibb’s Phenomenon, Parseval’s relation for power signals
9
Power density spectrum, Frequency spectrum.
3
Introduction to Fourier Series
The study of signals and systems using sinusoidal representation is termed as Fourier analysis.
❑ The representations are based on the periodicity and whether the given signal is a continuous time
signal or discrete time signal.
❑ All periodic signals have Fourier series representation.
❑ If the signal is periodic and continuous time signal, it will have continuous time Fourier series
representation.
❑ If the given signal is periodic and discrete time signal, it will have discrete time Fourier series
representation.
❑ All non periodic signals have Fourier transform representation.
Representation of Continuous time Periodic
signals
Trigonometric Fourier series equations
Practice Problems
2. Find the Trigonometric series coefficients for the figure given
3.
we get
Sub a0, an, bn values we get
4.
5. Find the Trigonometric series coefficients for the figure given
6. Find the Fourier series representation (trigonometric form) of the signal
Substituting a0, an, bn in x(t) equation
Trigonometric Fourier Series for Even and Odd Signals
Fourier Series Cosine Representation
The trigonometric Fourier series contains sine and cosine terms of the same frequency. By using
trigonometric identity, we can write
Thus we can obtain cosine representation of x(t) which contains sinusoids of frequencies Ωo,
2Ωo,…… . That is
Where
An-Amplitude coefficient
θn-Phase coefficient
1. Find the cosine representation Fourier series for the signal shown in Figure
2. Find the cosine Fourier series of half wave rectified function
The interval of integration is from t0 = 0 to t0+T = 2π. But during the interval from π to 2π
the signal has zero value
Symmetry Conditions
Symmetry Conditions
Application of symmetry conditions reduces the complexity in finding Fourier series or
Fourier series coefficients. If we have the knowledge about the symmetric condition of the
given signal, then we can directly calculate some coefficients.
The signal x(t) cand be splitted into even and odd functions
The even and odd parts of the signal x(t) can be obtained from the following relations
These relations can be used to find Fourier coefficients. To have convenient form we choose the
interval of integration –T/2 to T/2. Now
can be written as
can be written as
Substitute x(t) in new coefficients equations
If x(t) is even , then Xo(t)=0 then
Then an
Applying this to bn equation
Half Wave Symmetry
Problem:
Properties Of Continuous Time Fourier
Series
Proof for Properties Of Continuous Time Fourier Series
Gibb’s Phenomenon
Parseval’s Relation for Power Signals
Consider two periodic signals 𝑥1 (t) and 𝑥2(t) with equal periods 𝑇.
If the Fourier series coefficents of these two signals are 𝑐𝑛 and 𝑑𝑛 then
If 𝑥1(𝑡)= 𝑥2 (𝑡)= x(𝑡) then
The above equation can also be written as follows
Power density spectrum
Frequency Spectrum
Fourier spectrum of a periodic signal x(t) can be obtained by plotting the Fourier
coefficients versus Ω.
The plot of amplitude of Fourier coefficients 𝑐𝑛 versus Ω is known as amplitude spectra
and the plot of phase of Fourier coefficients උ𝑐𝑛versus Ω is known as phase spectra.
The two plots together are known as Fourier frequency spectra of x(t).
This type of representation is also known as frequency domain representation of x(t).
The spectrum exists only at discrete frequencies nΩ0, where n=0,1,2…
Thus the Fourier spectrum is s discrete spectrum otherwise called line spectrum.
1.
2.