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Fourier representation of signals

Pouyan Ebrahimbabaie

Laboratory for Signal and Image Exploitation (INTELSIG) Dept. of Electrical Engineering and Computer Science

University of Liège

Liège, Belgium

Applied digital signal processing (ELEN0071-1)

19 February 2020

MATLABtutorial series (Part 1.1)

Contacts

Email: P.Ebrahimbabaie@ulg.ac.be

Office: R81a

Tel: +32 (0) 436 66 37 53

Web: aie/ 2

Fourier analysis is like a glass prism

Glass prism

Analysis

Beam of

sunlight

VioletBlueGreenYellowOrangeRed

Fourier analysis is like a glass prism

4

Glass prism

Analysis

Beam of

sunlight

VioletBlueGreenYellowOrangeRed

Beam of

sunlightSynthesis

White light

Fourier analysis in signal processing

Fourier analysis is the decomposition of a signal into frequency components, that is, complex exponentials or sinusoidal signals.

Original signal

Fourier analysis in signal processing

Fourier analysis is the decomposition of a signal into frequency components, that is, complex exponentials or sinusoidal signals.

Sinusoidal signals

Original signal=

Joseph Fourier

1768-1830

Motivation

Question:what is our motivation to describe each signal as a sum or integral of sinusoidal signals?

Motivation

Question:what is our motivation to describe each signal as a sum or integral of sinusoidal signals? Answer: the major justification is that LTI systems have a simple behavior with sinusoidal inputs. Notice:the response of a LTI system to a sinusoidal is sinusoid with the same frequency but different amplitude and phase.

Motivation

Question:what is our motivation to describe each signal as a sum or integral of sinusoidal signals? Answer: the major justification is that LTI systems have a simple behavior with sinusoidal inputs. Interesting application: we can remove selectivelya

Notations and abbreviations

Mathematical tools for frequency analysis depends on,

Nature of time: continuous or discrete

Existence of harmonic: periodic or aperiodic

Notations and abbreviations

Mathematical tools for frequency analysis depends on,

Nature of time: continuous or discrete

Existence of harmonic: periodic or aperiodic

The signal could be,

Continuous-time andperiodic

Continuous-time andaperiodic

Discrete-time andperiodic

Discrete-time andaperiodic

Notations and abbreviations

Mathematical tools for frequency analysis depends on,

Nature of time: continuous or discrete

Existence of harmonic: periodic or aperiodic

The signal could be,

Continuous-time andperiodic (freq. dom. CTFS)

Continuous-time andaperiodic (freq. dom. CTFT)

Discrete-time andperiodic (freq. dom.DTFS)

Discrete-time andaperiodic (freq. dom. DTFT)

Notations and abbreviations

Mathematical tools for frequency analysis depends on,

Nature of time: continuous or discrete

Existence of harmonic: periodic or aperiodic

The signal could be,

Continuous-time and periodic(freq. dom. CTFS)

Continuous-time and aperiodic (freq. dom. CTFT)

Discrete-time and periodic(freq. dom. DTFS)

Discrete-time and aperiodic (freq. dom. DTFT)

Notice: when the signal is periodic, we talk about

Fourier series (FS).

Notations and abbreviations

Mathematical tools for frequency analysis depends on,

Nature of time: continuous or discrete

Existence of harmonic: periodic or aperiodic

The signal could be,

Continuous-time and periodic (freq. dom. CTFS)

Continuous-time and aperiodic(freq. dom. CTFT)

Discrete-time and periodic (freq. dom. DTFS)

Discrete-time and aperiodic (freq. dom. DTFT)

Notice: when the signal is aperiodic, we talk about

Fourier transform (FT).

Continuous-time periodic signal: CTFS

2pW0T0

Continuous-time signals

x(t) 0

Time-domainFrequency-domain

Continuous and periodicDiscrete and aperiodic

t0 ck

W-T00T

Continuous-time periodic signal: CTFS

2pW0T0

Continuous-time signals

x(t) 0

Time-domainFrequency-domain

Continuous and periodicDiscrete and aperiodic

t0 ck

W-T00T

From CTFS to CTFT

Example: consider the following signal,

From CTFS to CTFT

Example: consider the following signal,

From CTFS to CTFT

From CTFS to CTFT

From CTFS to CTFT

From CTFS to CTFT

Continuous-time aperiodic signal: CTFT

Continuous-time aperiodic signal: CTFT

Continuous-time aperiodic signal: CTFT

Discrete-time periodic signal: DTFS

Discrete-time signals

x[n]ck -NN0

Discrete and periodicDiscrete and periodic

nk

Time-domainFrequency-domain

-NN0

Discrete-time periodic signal: DTFS

Discrete-time signals

x[n]ck -NN0

Discrete and periodicDiscrete and periodic

nk

Time-domainFrequency-domain

-NN0

Discrete-time aperiodic signal: DTFT

Discrete-time signals

X(ejw)

-4-2204-2p-p0p2p

Continous and periodic

nw

Time-domainFrequency-domain

Discrete and aperiodic

x[n]

Discrete-time aperiodic signal: DTFT

Discrete-time signals

X(ejw)

-4-2204-2p-p0p2p

Continous and periodic

nw

Time-domainFrequency-domain

Discrete and aperiodic

x[n]

Discrete-time aperiodic signal: DTFT

Discrete-time signals

X(ejw)

-4-2204-2p-p0p2p

Continous and periodic

nw

Time-domainFrequency-domain

Discrete and aperiodic

x[n]

Everything you need to know !

31

Summary of Fourier series and transforms

32
ࢻin one domain implies discretization domain, and vice versa.

Frequency : F(Hz)

Angular frequency: ષൌ૛࣊ࡲ(rad/sec) Normalized frequency: fൌࡲȀࡲ࢙(cycles/samples)

Normalized angular frequency:

࣓ൌ૛࣊ൈࡲȀࡲ࢙(radians x cycles/samples) radians

Normalized angular frequency:

࣓ൌ૛࣊ൈࡲȀࡲ࢙(radians x cycles/samples) radians

Low Freq.High Freq.High Freq.

Numerical computation of DTFS

FormulaMATLABfunction

includes first ࡺsampls.

Example 1.1: use of fft and ifft

Example 1: Compute the DFTS of pulse train with ࡸ=2 and ࡺൌ % signal x=[1 1 1 0 0 0 0 0 1 1] % Nquotesdbs_dbs6.pdfusesText_11