# Lecture04 by rite2methun

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

Miroslav Lutovac and Dejan Tosic

Overview
• • • • • • 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

Definition
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

Properties
Uniqueness

Homogeneity

Differentiation
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)
H(jw)

Y ( jw)

Frequency response

Discrete Fourier transform (DFT)
and FFT

Definition

Finite-length sequence

transformation

DFT

Inverse discrete Fourier transform (IDFT)

Discrete Fourier transform pair

FFT

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
j

Normalized frequency (MATLAB)

Frequency response

MATLAB freqz