Lecture04 by rite2methun


									Analog and Digital Filter Design Fourier series, transform, and FFT

Miroslav Lutovac and Dejan Tosic

• • • • • • Fourier series Fourier transform FFT Examples MATLAB fft MATLAB freqz

Fourier series

Periodic signals
• Nonsinusoidal periodic signals play an important part in signal processing • Theory of nonsinusoidal periodic signals is based upon resolving them into sinusoidal components • According to the principle of superposition, we can find the steady-state response of an LTI system to an arbitrary periodic input by applying the phasor method to harmonic components

Sectionally continuous signal
• Consider a real signal x(t) of real variable t that satisfies the following conditions: • x(t+T) = x(t); that is, the signal is periodic having a period T • x(t) is defined in the interval t < t < t + T • x(t) and dx(t)/dt are sectionally continuous in t < t < t + T

Definition of Fourier series

fundamental angular frequency

Fourier coefficients The process of resolving a signal into its Fourier series is called spectral analysis or harmonic analysis

Complex Form of Fourier Series

Parseval's Identity

If the signal x(t) represents an electric current or voltage across a resistor, the expression is proportional to the resistor average power over a period

Harmonics of periodic signals

dc component

ac component nth harmonic

Gibbs phenomenon
When a sudden change of amplitude occurs in a signal and the attempt is made to represent it by a finite number of terms in a Fourier series, the overshoot at the corners (at the points of abrupt change) is always found. As the number of terms is increased, the overshoot is still found; this is called the Gibbs phenomenon.

Amplitude, phase, power spectrum
A graphical representation of a periodic signal x(t) produced by drawing a series of vertical lines at intervals on a horizontal axis, where the intervals represent an increase of n

amplitude spectrum

phase spectrum

power spectrum

Example spectrum

Fourier transform

Fourier transform

Inverse Fourier transform

The variable w is called a continuous frequency variable

sufficient existence condition

Spectrum of x(t)
Plots of |X(jw)| and arg(X(jw)) versus w are called the amplitude spectrum and phase spectrum of x(t), respectively




Dx(t )  dx(t ) dt

Parseval's theorem and energy spectral density

Total energy

Energy spectral density

Frequency response of continuous-time systems
x(t ) y(t )

Relaxed LTI system described by linear constant-coefficients differential equation

Y ( jw ) H ( jw )  X ( jw )
X ( jw)

Y ( jw)

Frequency response

Discrete Fourier transform (DFT)
and FFT


Finite-length sequence



Inverse discrete Fourier transform (IDFT)

Discrete Fourier transform pair


Parseval's identity

Frequency response of discrete-time systems
{x(n) } { y(n) }

Relaxed LTI system described by linear constant-coefficients difference equation

The system transforms the input sequence by multiplying each member of the input DFT with the factor H(k)

Digital frequency
k   2 N
Digital angular frequency

 k  2 N

Digital frequency

 k 2  N
H (e )  H ( z) z e j

Normalized frequency (MATLAB)

Frequency response

MATLAB freqz

Further reading

M. D. Lutovac, D. V. Tošić, B. L. Evans
Filter Design for Signal Processing Using MATLAB and Mathematica Prentice Hall Upper Saddle River, New Jersey ISBN 0-201-36130-2, (c) 2001 http://kondor.etf.bg.ac.yu/~lutovac/

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