3. Fourier Series and Transform

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 1/71

BITS Pilani, K K Birla Goa Campus

Find the response of an LTI system by expressing the

excitation as a linear combination of impulses and

expressing the response as a linear combination of 

impulse responses.

We called that technique convolution.

In earlier lectures…. 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 2/71

BITS Pilani, K K Birla Goa Campus

Sinusoidal Signal 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 3/71

BITS Pilani, K K Birla Goa Campus

Representing Real Signals using Complex

Phasors 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 4/71

BITS Pilani, K K Birla Goa Campus

Advantages1. Simplifies mathematical derivations and

analyses.

2. Addition of signals is easy

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 5/71

BITS Pilani, K K Birla Goa Campus

Fourier series/Transform

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 6/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 7/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 8/71BITS Pilani, K K Birla Goa Campus

Time property of signal and itsFourier representations

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 9/71BITS Pilani, K K Birla Goa Campus

• Fourier Series is an alternate method of signal

representation.

• Signal is represented as a function of 

frequency.

• Every Periodic Signal (real world signal) can

be expressed in terms of its Fourier

coefficients.

• If the Fourier coefficients are known it is

possible to synthesize the periodic signal

Fourier series

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 10/71BITS Pilani, K K Birla Goa Campus

Periodic Signal 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 11/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 12/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 13/71BITS Pilani, K K Birla Goa Campus

Trigonometric Form

Exponential Form

Fourier series representation

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 14/71BITS Pilani, K K Birla Goa Campus

Trigonometric Form 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 15/71BITS Pilani, K K Birla Goa Campus

Compact Trigonometric Form 

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 16/71BITS Pilani, K K Birla Goa Campus

Example

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 17/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 18/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 19/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 20/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 21/71BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 22/71BITS Pilani, K K Birla Goa Campus

Example 2

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 23/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 24/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 25/71

BITS Pilani, K K Birla Goa Campus

Exponential Form

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 26/71

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 27/71

BITS Pilani, K K Birla Goa Campus

Example

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 28/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 29/71

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 30/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 31/71

BITS Pilani, K K Birla Goa Campus

Periodic symmetries & Fouriercoefficients

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 32/71

BITS Pilani, K K Birla Goa Campus

Fourier coefficients

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 33/71

BITS Pilani, K K Birla Goa Campus

Line Spectrum

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 34/71

BITS Pilani, K K Birla Goa Campus

2-Sided Line Spectrum

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 35/71

BITS Pilani, K K Birla Goa Campus

P S f i di

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 36/71

BITS Pilani, K K Birla Goa Campus

Power Spectrum of a periodicsignal

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 37/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 38/71

BITS Pilani, K K Birla Goa Campus

• It is the difference between the highest and lowestfrequencies of the spectral component of a signal.

Bandwidth of the signal

D t i ti f f d t l

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 39/71

BITS Pilani, K K Birla Goa Campus

Determination of fundamentalfrequency and period

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 40/71

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 41/71

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 42/71

BITS Pilani, K K Birla Goa Campus

Transforms a signal (i.e., function) from the spatialdomain to the frequency domain.

If the signal is aperiodic signal, Fourier Transform isused.

Fourier Transform

An aperiodic function is a function that never repeats itself 

Never Repeating itself is equivalent to Repeating at Infinity. 

=> Aperiodic Function spans a Continuum of Frequencies of 

the Fourier Series Envelope

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 43/71

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 44/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 45/71

BITS Pilani, K K Birla Goa Campus

In Fourier series,

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 46/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 47/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 48/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 49/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 50/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 51/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 52/71

BITS Pilani, K K Birla Goa Campus

Examples

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 53/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 54/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 55/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 56/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 57/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 58/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 59/71

BITS Pilani, K K Birla Goa Campus

Dirichlet’s conditions

Fourier Transform of basic

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 60/71

BITS Pilani, K K Birla Goa Campus

Fourier Transform of basicfunctions

Impulse function ----- Fourier Transform

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 61/71

BITS Pilani, K K Birla Goa Campus

Inverse Fourier Transform

The Fourier transform and its inverse are referred

to as a Fourier transform pair

Inverse Fourier tranform of

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 62/71

BITS Pilani, K K Birla Goa Campus

Inverse Fourier tranform ofImpulse function

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 63/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 64/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 65/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 66/71

BITS Pilani, K K Birla Goa Campus

R f th t

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 67/71

BITS Pilani, K K Birla Goa Campus

Response of the system

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 68/71

BITS Pilani, K K Birla Goa Campus

P i f F i T f

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 69/71

BITS Pilani, K K Birla Goa Campus

Properties of Fourier Transform

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 70/71

BITS Pilani, K K Birla Goa Campus

8/2/2019 3. Fourier Series and Transform

http://slidepdf.com/reader/full/3-fourier-series-and-transform 71/71