Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Lecture 1

Frequency Analysis ofDiscrete-time Signals

(4.2.1-4.2.3,4.2.5)

Dr D. Laurenson

27th May, 2013

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Course structure

Fourier transforms (3 lectures)

Correlation (2 lectures)

Linear systems (1 lecture)

Class test (1 class)

Random signals (2 lectures)

Digital ˛lters (2 lectures)

Power spectral estimation (2 lectures)

Multirate signal processing (1 lecture)

Analogue to digital converter (1 lecture)

Revision (1 lecture)

Matched ˛lters (2 lectures)

Wiener ˛lters (2 lectures)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Outline

1 Introduction

2 Periodic signals

3 Power Density Spectra

4 Aperiodic Signals

5 Energy Density Spectrum

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Aperiodic, continuous time signals

Aperiodic in time implies continuous in frequency,thus the Fourier transform is used:

X(F ) =

Z 1`1

x(t)e`j2ıFtdt (4.1.30)

or, equivalently,

X(!) =

Z 1`1

x(t)e`j!tdt

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Aperiodic, continuous time signals

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Periodic, continuous time signals

Periodic in time implies sampled in frequency, thusthe Fourier Series is used. The sample spacing isF0 = 1

TPHz for a period TP s

ck =1

TP

ZTp

x(t)e`j2ıkF0tdt (4.1.9)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Continuous time signals

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete time periodic signals

Aperiodic in time ) Continuous in frequency

Periodic in time ) Sampled in frequency

By analogy, Sampled in time ) Periodic infrequency

Frequency domain is often normalised

Denote sampling frequency as 2ı

For a signal that is periodic in time, withperiod N, the sample spacing in frequency is ´t

2ı

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete-time Fourier Series

The Discrete-time Fourier Series (DTFS) is de˛nedby

ck =1

N

N`1Xn=0

x(n)e`j2ıkn=N (4.2.8)

x(n) =

N`1Xk=0

ckej2ıkn=N (4.2.7)

Sometimes the normalisation of 1N

is performed inthe synthesis equation (4.2.7) instead of theanalysis (4.2.8)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

DTFS example

x(n) = cos(ın=3) ) N = 6

) ck =1

N

N`1Xn=0

x(n)e`j2ıkn=N

=1

6

5Xn=0

cos(ın=3)e`j2ıkn=6

=1

6

5Xn=0

1

2

nejın=3 + e`jın=3

oe`j2ıkn=6

=1

12

5Xn=0

nej2ı(1`k)n=6 + e`j2ı(1+k)n=6

o

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

DTFS example

ck =1

12

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

5Xn=0

ej2ı(1`k)n=6 +

5Xn=0

1

if k = `1

5Xn=0

1 +

5Xn=0

e`j2ı(1+k)n=6

if k = 1

5Xn=0

ej2ı(1`k)n=6 +

5Xn=0

e`j2ı(1+k)n=6

otherwise

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

DTFS example

N`1Xn=0

ej2ıkn=N =

N`1Xn=0

“ej2ık=N

”n

=1`

`ej2ık=N

´N1`

`ej2ık=N

´= 0 8k =2 (0;˚N;˚2N; : : :)

and 8k 2 (0;˚N;˚2N; : : :)

N`1Xn=0

ej2ıkn=N =

N`1Xn=0

1 = N

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

DTFS example

) ck =1

2

(1 ; k = `1 or k = 1

0 ; otherwise

Thus the DTFS of a sampled cosine is periodic,with period N, and consists of two non-zerofrequency components representing the twocomplex phasors that construct the cosine

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Power Density Spectra

DTFS is a transform

Relative delay is represented by the phasecomponent

Power Density Spectra represent the power ofthe constituent frequency components

The derivation of the PDS starts with thepower of a signal:

Px =1

N

N`1Xn=0

jx(n)j2 (4.2.10)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Power Density Spectrum de˛nition

Expanding this, and representing x(n) by itsDTFS:

Px =1

N

N`1Xn=0

0@N`1Xk=0

ckej2ıkn=N

1Ax˜(n)

Switching the order of summation:

Px=

N`1Xk=0

ck

0@ 1

N

N`1Xn=0

x˜(n)ej2ıkn=N

1A=

N`1Xk=0

ckc˜k

jckj2 is the Power Density Spectrum

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

PDS example

x(n) =

(A ; 0 » n < L

0 ;L » n < N

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

PDS example

ck =1

N

N`1Xn=0

x(n)e`j2ıkn=N

=1

N

L`1Xn=0

A:“e`j2ık=N

”nM`1Xm=0

gn =1` gM

1` g

) ck =A

N

1` e`j2ıkL=N

1` e`j2ık=N

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

PDS example

jckj2 =A2

N2´ 1` e`j2ıkL=N

1` e`j2ık=N´ 1` ej2ıkL=N

1` ej2ık=N

=A2

N2´ 2` e`j2ıkL=N ` ej2ıkL=N

2` e`j2ık=N ` ej2ık=N

=A2

N2´``ej2ıkL=2N ` e`j2ıkL=2N

´2

``ej2ık=2N ` e`j2ık=2N

´2

=A2

N2´

sin2(ıkL=N)

sin2(ık=N)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

PDS example

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Cycles/Sampling interval)

N |

ck |

L = 5, N = 10

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

PDS example

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Cycles/Sampling interval)

N |

ck |

L = 5, N = 40

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Power Density Spectrum properties

If a periodic sampled signal is real:

By de˛nition, x˜(n) = x(n)

c˜`k = ck, and c˜N`k = ck

jc`kj = jckj, and jcN`kj = jckj](c`k) = `](ck), and ](cN`k) = `](ck)

<(c`k) = <(ck) and =(c`k) = `=(ck)

<(cN`k) = <(ck) and =(cN`k) = `=(ck)

So for the PDS: jckj2 = jc`kj2 = jcN`kj2.

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete-time Aperiodic Signals

Aperiodic signals have a continuous frequencytransform

Discrete-time (i.e. sampled) signals have aperiodic frequency transform

X(!) =

1Xn=`1

x(n)e`j!n (4.2.23)

This transform is also known as thediscrete-time Fourier transform (DTFT)

The period is fs = 1´t

, or !s = 2ı´t

The normalised frequency response is de˛nedby ´t = 1, giving rise to:

X(! + 2ık) = X(!) (4.2.24)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete-time Fourier TransformExample

Let the signal be an aperiodic rectangular pulse:

x(n) =

(A ; 0 » n < L

0 ;n < 0 or n – L

X(!) =

1Xn=`1

x(n)e`j!n

=

L`1Xn=0

Ae`j!n = A1` e`j!L

1` e`j!

= A ´ e`j!L=2

e`j!=2´ e

j!L=2 ` e`j!L=2

ej!=2 ` e`j!=2

= Ae`j!(L`1)=2 ´sin(!L=2)

sin(!=2)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete-time Fourier TransformExample

0

A

-3π -2π -π 0 π 2π 3π

Mag

nitu

de

Frequency (radians/sampling interval)

-π

0

π

-3π -2π -π 0 π 2π 3π

Pha

se

Frequency (radians/sampling interval)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Discrete-time Aperiodic Signals

The reverse transform is given by

x(n) =1

2ı

Z ı

`ıX(!)ej!nd! (4.2.27)

This may be computed over any full period ofX(!), thus may also be written as:

x(n) =1

2ı

Z 2ı

0

X(!)ej!nd!

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Energy Density Spectrum

The energy density spectrum describes energy in asignal in terms of its frequency response:

Ex =

1Xn=`1

jx(n)j2 =1

2ı

Z ı

`ıjX(!)j2 d!

(4.2.41)The energy density spectrum is the integrand andde˛ned as:

Sxx(!) = jX(!)j2

For real valued signals,Sxx(`!) = Sxx(!) = Sxx(2ı ` !)

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Energy Density Spectrum Example

Let x(n) be de˛ned as:

x(n) =

(an ;n – 0

0 ;n < 0

where `1 < a < 1.

X(!) =

1Xn=`1

x(n)e`jn! =

1Xn=0

`ae`j!

´n

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Energy Density Spectrum Example

X(!) =1

1` ae`j!

) Sxx(!) =1

1` ae`j!1

1` aej!

=1

1 + a2 ` a (ej! + e`j!)

=1

1` 2a cos(!) + a2

The resulting spectrum depends upon theparameter a. For a = 0, which results in the inputbeing a single non-zero sample, the spectrum is‚at.

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Energy Density Spectrum Example

Lecture 1

Introduction

Periodicsignals

PowerDensitySpectra

AperiodicSignals

EnergyDensitySpectrum

Summary

1 Introduction

2 Periodic signals

3 Power Density Spectra

4 Aperiodic Signals

5 Energy Density Spectrum