2. spectral analysis of discrete signals and systems€¦ · spectral analysis of discrete signals...

2. Spectral Analysis of Discrete Signals andSystems

in: Signal Processing for Spatial Sound Control

Dr.-Ing. Gerald Enzner

Institut für Kommunikationsakustik (IKA)

Winter Term, 2010/2011

2. Spectral Analysis

2.1 The Fourier Transform (FT)

2.2 The Fourier Transform of Discrete Signals (DSFT)

2.3 Frequency Response of Discrete Systems

2.4 The Discrete Fourier Transform (DFT,FFT)

2.5 Time-Frequency Representation of Audio Signals

Dr.-Ing. Gerald Enzner • Ruhr-Universität Bochum 2/28

2.1 Fourier Transform

The Fourier transform (FT) provides an analysis of signals in terms of itsharmonic components. The FT relates a continuous (time) domain signalxa(t) to its Fourier domain representation Xa(jω)

Xa(jω) =

Z ∞

−∞

xa(t)e−jωtdt

as a function of the angular frequency ω = 2πf . The inverse relationship isgiven by

xa(t) =1

2π

Z ∞

−∞

Xa(jω)ejωtdω .

For signals xa(t) with infinite support, the FT exists if the conditionZ ∞

−∞

|xa(t)|dt < ∞

holds. The inverse FT then reconstructs a signal which is identical to theoriginal signal except for a finite number of discontinuities. Xa(jω) is ingeneral a continuous and non-periodic function of frequency.


FT of a Rectangular Pulse

x(t)= rect( t

T

)

=

1 |t| ≤ T/2

0 |t| > t−−−−• X(jω)=T si

(

ωT/2)

=Tsin(ωT/2)

ωT/2

Example: xa(t) = rect(

tT

)with T = 60/1000 s

−0.05 0 0.05−0.2

0

0.2

0.4

0.6

0.8

1

t / s

x(t)

−400 −200 0 200 400−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ω / (2 π rad)

X(jω

)


Properties of the Fourier Transform

property time domain frequency domain

transform x(t) =1

2π

Z ∞

−∞

X(jω)ejωtdω X(jω) =

Z ∞

−∞

x(t)e−jωtdt

linearity ax1(t) + bx2(t) aX1(jω) + bX2(jω)

conjugation x∗(t) X∗(−jω)

symmetry x(t) is a real valued signal X(−jω) = X∗(jω)

even part xe(t) = 0.5(x(t) + x(−t)) ReX(jω)

odd part xo(t) = 0.5(x(t) − x(−t)) jImX(jω)

convolution x1(t) ∗ x2(t) X1(jω) · X2(jω)

time shift x(t − t0) e−jωt0 X(jω)

modulation x(t)ejωM t X(j(ω − ωM))

scaling x(at), a ∈ R, a 6= 01|a|

X“

jω

a

”

Parseval’s theorem

Z ∞

−∞

|x(t)|2dt =1

2π

Z ∞

−∞

|X(jω)|2dω


2.2 Fourier Transform of Discrete Signals

The Fourier transform of discrete signals (DSFT) is derived using therepresentation of sampled signals as pulse trains,

xs(t) =

∞∑

k=−∞

xa(kT)δa(t − kT) ,

and using the sifting property of the Dirac impulse, i.e.,∫

∞

−∞

δa(t − b)f (t)dt = f (b) .

For a sampling period of T =1fs

, we obtain

Xs(jω) =

∫∞

−∞

xs(t)e−jωtdt =

∞∑

k=−∞

xa(kT)

∫∞

−∞

δ(t − kT)e−jωtdt

=

∞∑

k=−∞

xa(kT)e−jωkT .


Fourier Transform of Discrete Signals

Xs(jω) is a continuous and periodic function of the angular frequencyω and hence also of frequency f . To see this we note that thecomplex phasor

e−jωkT = e−j2πfkT = cos(2πfkT) − j sin(2πfkT)

is periodic in ω with period ω =2π

T= 2πfs. Therefore,

Xs(j(ω +2πℓ

T)) = Xs(jω) for any ℓ ∈ Z .

In digital signal processing, we treat a sampled signal xs(t) on anabstract level as a sequence of numbers x(k). This allows a generaltreatment of signal processing methods without reference to thephysical aspects (time or frequency scale) of the signal. Occasionally,we remind ourselves that the sequence x(k) was obtained bysampling a continuous-time signal xa(t) with sampling rate fs.


Fourier Transform of Discrete Signals

To facilitate the treatment of sampled signals in the Fourier domain,we normalize the frequency variable f on the sampling rate fs andintroduce the normalized angular frequency Ω = ωT = 2πfT = 2π f

fs.

We then obtain the Fourier Transform of Discrete Signals (DSFT) andits inverse relationship

X(ejΩ) =

∞∑

k=−∞

x(k)e−jΩk x(k) =1

2π

∫ π

−π

X(ejΩ)ejΩkdΩ .

Note, that the inverse transform is evaluated over one period of thespectrum only. X(ejΩ) is a complex quantity:

X(ejΩ) = ReX(ejΩ) + jImX(ejΩ) = |X(ejΩ)|ejφ(Ω)

where |X(ejΩ)| is the amplitude spectrum and φ(Ω) ∈ −π, π is thephase. Frequently, we consider the logarithm of the amplitudespectrum 20 log10(|X(ejΩ)|) = |X(ejΩ)|/dB.


Comparison: FT and DSFT of a Rectangular Pulse

−0.05 0 0.05−0.2

0

0.2

0.4

0.6

0.8

1

t / s

x(t)

−400 −200 0 200 400−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ω / (2 π rad)

X(jω

)

−10 −5 0 5 10−0.2

0

0.2

0.4

0.6

0.8

1

k

x(k)

−2 −1 0 1 2−4

−2

0

2

4

6

8

10

12

14

Ω / (2 π)

X(e

jΩ)

∑(N−1)/2ℓ=−(N−1)/2 δ (k − ℓ) −−−−• sin(ΩN/2)

sin(Ω/2)


Properties of the DSFT


transform x(k) =1

2π

Z

π

−π

X(ejΩ)ejΩkdΩ X(ejΩ) =∞

X

k=−∞

x(k)e−jΩk

linearity ax1(k) + bx2(k) aX1(ejΩ) + bX2(ejΩ)

conjugation x∗(k) X∗(e−jΩ)

symmetry x(k) is a real valued signal X(e−jΩ) = X∗(ejΩ)

even part xe(k) = 0.5(x(k) + x(−k)) ReX(ejΩ)

odd part xo(k) = 0.5(x(k) − x(−k)) jImX(ejΩ)

convolution x1(k) ∗ x2(k) X1(ejΩ) · X2(ejΩ)

time shift x(k − k0) e−jΩk0X(ejΩ)

modulation x(k)ejΩM k X(ej(Ω−ΩM))

Parseval’s theorem∞

X

k=−∞

x(k)y∗(k) =1

2π

Z

π

−π

X(ejΩ)Y∗(ejΩ)dΩ


Example: DSFTof Exponential x(k) = aku(k), a = 0.7

−1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

Ω / (2 π rad)

Re

X(e

jΩ)

−1.5 −1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Ω / (2 π rad)

ImX

(ejΩ

)

−1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

Ω / (2 π rad)

|X(e

jΩ)|

−1.5 −1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Ω / (2 π rad)

φ(Ω

)


2.3 Frequency Response of Discrete Systems

impulse response

frequency response

x(k) y(k) = x(k) ∗ h(k)

h(k)

X(ejΩ) Y(ejΩ) = X(ejΩ) · H(ejΩ)

H(ejΩ)


Example: Delay System

h(k) = δ(k − k0) −−−−• H(ejΩ) =∞∑

k=−∞δ(k − k0)e−jΩk = e−jΩk0

0 0.2 0.4 0.6 0.8 1−2000

−1500

−1000

−500

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1


Mag

nitu

de (

dB)

k0 = 10


Example: Non-Recursive (FIR) Smoothing

0 20 40 60 80 100−4

−3

−2

−1

0

1

2

3

4

k

x(k)

0 20 40 60 80 100−3

−2

−1

0

1

2

3

4

k

y(k)

N = 2

0 20 40 60 80 100−2

−1

0

1

2

3

k

N = 4

y(k)

0 20 40 60 80 100−1

−0.5

0

0.5

1

1.5

k

y(k)

N = 16


Frequency Response of FIR Smoothing Filter

0 0.2 0.4 0.6 0.8 1−200

−100

0

100


Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1−60

−40

−20

0


Mag

nitu

de (

dB)

N = 16


Example: First Order Recursive (IIR) Smoothing

0 20 40 60 80 100−4

−3

−2

−1

0

1

2

3

4

k

x(k)

0 20 40 60 80 100−3

−2

−1

0

1

2

3

4

k

y(k)

α = 0.333

0 20 40 60 80 100−2

−1

0

1

2

3

k

y(k)

α = 0.6

0 20 40 60 80 100−1

−0.5

0

0.5

1

1.5

k

y(k)

α = 0.8824


Frequency Response of IIR Smoothing Filter

0 0.2 0.4 0.6 0.8 1−80

−60

−40

−20

0


Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1−30

−20

−10

0


Mag

nitu

de (

dB)

α = 0.8824


2.4 Definition of the DFT

The Discrete Fourier Transform (DFT) is given by

Xµ =

M−1∑

k=0

x(k)e−j 2πµkM , µ = 0, . . . , M − 1

and the inverse relationship by

x(k) =1M

M−1∑

µ=0

Xµej 2πµkM , µ = 0, . . . , M − 1 .

The coefficients of the DFT are spaced by ∆Ω = 2πM on the

normalized frequency axis Ω. Therefore, we may think of the DFT asan approximation of a continuous spectrum at discrete frequenciesΩµ = 2πµ

M . When the signal samples x(k) are generated by means ofsampling a continuous signal x(t) with sampling period T = 1

fs, the

coefficients of the DFT are spaced as fµ = µfsM , therefore ∆f = fs

M .


Properties of the DFT


transform x(k) =1M

M−1X

µ=0

Xµej2πµkM Xµ =

M−1X

k=0

x(k)e−j2πµkM

k = 0 . . . M − 1 µ = 0 . . . M − 1

linearity ax(k) + by(k) aXµ + bYµ

symmetry x(k) is real-valued Xµ = X∗[−µ]modM

convolution (x(k)RM(k) ⊛ y(k)RM(k)) XµYµ

multiplication x(k)y(k)1M

(XµRM(µ) ⊛ YµRM(µ))

delay x([k + k0]modM) e+j2πµk0

M Xµ

modulation x(k)e−j2πkµ0

MX[µ+µ0 ]modM

Parseval’stheorem

M−1X

k=0

|x(k)|2 =1M

M−1X

µ=0

|Xµ|2

RM (k) denotes a rectangular window with RM (k) = 1 for 0 ≤ k ≤ M − 1 and RM (k) = 0 otherwise.


2.5 Zeit-Frequenz Darstellung von Audiosignalen

0 1 2 3 4 5 6 7 8 9−1

−0.5

0

0.5

1

Am

pliu

tude

0 1 2 3 4 5 6 7 8 9−1

−0.5

0

0.5

1

Zeit [Sekunden]

Am

pliu

tude

Sprache

Musik


Segmentelle Kurzzeit-Spektralanalyse

......abs(DFT()) abs(DFT())...

... ...

Zeit [Sekunden]

Freq

uenz

[Her

tz]

0.5 1 1.5 2 2.50

5000

10000

15000

−100

−50

0

0 0.5 1 1.5 2 2.5−1

0

1

Zeit [Sekunden]A

mpl

iutu

de

Spectrogramm

Zeitsignal


Audio SpektrogrammeF

requ

enz

[Her

tz]

1 2 3 4 5 6 7 8 90

5000

10000

15000

Zeit [Sekunden]

Fre

quen

z [H

ertz

]

1 2 3 4 5 6 7 8 90

5000

10000

15000

−100

−50

0

−100

−50

0

Sprache

Musik


Zeit-Bandbreite UnschärfeF

requ

enz

[Her

tz]

0.5 1 1.5 2 2.50

5000

10000

15000

Zeit [Sekunden]

Fre

quen

z [H

ertz

]

0.5 1 1.5 2 2.50

5000

10000

15000

−100

−50

0

−150

−100

−50

0

Hohe Spektrale Auflösung (32 ms Fensterung)

Hohe Zeitliche Auflösung (8 ms Fensterung)


Systemtheorie der Kurzzeit-Spektralanalyse

x(k)

DFT

(FFT)

M

XM-1(k)

XM-2(k)

Xµ(k)

X1(k)

X0(k)

z-1

z-1

z-1

z-1

...

...

xk(M-2)

xk(M-1)

xk(κ)

xk(1)

xk(0)

...

...

h(0)

h(1)

h(M-1-κ)

h(M-2)

h(M-1)

...

...

x(k-M+1+κ)

x(k)

x(k-1)

x(k-M+2)

x(k-M+1)

...

...


DFT als Filterbank

Xµ(k) =M−1∑

κ=0

xk(κ)wµκM Def. DFT, wM=e−j 2π

M

=

M−1∑

κ=0

x(k − M + 1 + κ)h(M − 1 − κ)wµκM einsetzen (siehe Bild)

=

∞∑

κ=−∞

x(k − M + 1 + κ)h(M − 1 − κ)wµκM w(k) zeitlich begrenzt

=∞∑

u=−∞

x(u)h(k − u)w−µ(k−u−M+1)/M Subst.: k−M+1+κ=u

= x(k) ∗ [h(k)w−µ(k+1)M ]

︸︷︷︸

Bandpässe mit Mittenfrequenzen Ωµ=µ 2π

M

Faltung


Beispiel: Verrrauschte Sprache

0 0.5 1 1.5 2 2.5−1

0

1A

mpl

itude

Zeit [Sekunden]

Fre

quen

z [H

ertz

]

0.5 1 1.5 2 2.50

5000

10000

15000

−80−60−40−2002040

Zeitsignal

Spektrogramm


Zusammenfassung und Ausblick

Eigenschaften der Zeit-Frequenz Darstellung:

• Gleitende Kurzzeit-Spektralanalyse

• Reichhaltige Audiosignaldarstellung mit zeitlicher undspektraler Charakteristik

• Berechnung z.B. mit DFT oder expliziter Bandpaßanalyse(Filterbänke: gleichförmig, ungleichförmig)

• Grenzen: Zeit-Bandbreite Unschärfe


Zusammenfassung und Ausblick

Eigenschaften der Zeit-Frequenz Darstellung:

• Gleitende Kurzzeit-Spektralanalyse

• Reichhaltige Audiosignaldarstellung mit zeitlicher undspektraler Charakteristik

• Berechnung z.B. mit DFT oder expliziter Bandpaßanalyse(Filterbänke: gleichförmig, ungleichförmig)

• Grenzen: Zeit-Bandbreite Unschärfe

Anwendungsgebiete:

• Visualisierung zeitlicher und spektraler Charakteristik

• Spektrale Codierung und Übertragung von Audiosignalen

• Spektrale Methoden zur Filterung von Audiosignalen, z.B. fürQuellentrennung oder Störgeräuschreduktion


2. spectral analysis of discrete signals and systems€¦ · spectral analysis of discrete signals...

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