Chapter 7

Frequency Analysis of Signals and Systems

Nguyen Thanh Tuan, Click

M.Eng.

to edit Master subtitle style

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: nttbk97@yahoo.com

Frequency analysis of signal involves the resolution of the signal into

its frequency (sinusoidal) components. The process of obtaining the

spectrum of a given signal using the basic mathematical tools is

known as frequency or spectral analysis.

The term spectrum is used when referring the frequency content of a

signal.

The process of determining the spectrum of a signal in practice base

on actual measurements of signal is called spectrum estimation.

The instruments of software programs

used to obtain spectral estimate of such

signals are kwon as spectrum analyzers.

Digital Signal Processing

2

Frequency analysis of signals and systems

The frequency analysis of signals and systems have three major uses

in DSP:

1) The numerical computation of frequency spectrum of a signal.

2) The efficient implementation of convolution by the fast Fourier

transform (FFT)

3) The coding of waves, such as speech or pictures, for efficient

transmission and storage.

Digital Signal Processing

3

Frequency analysis of signals and systems

Content

1. Discrete time Fourier transform DTFT

2. Discrete Fourier transform DFT

3. Fast Fourier transform FFT

Digital Signal Processing

4

Transfer functions

and Digital Filter Realizations

1. Discrete-time Fourier transform (DTFT)

The Fourier transform of the finite-energy discrete-time signal x(n) is

defined as:

X ( ) x(n)e jn

n

where ω=2πf/fs

The spectrum X(w) is in general a complex-valued function of

frequency:

X ( ) | X () | e j ( )

where () arg( X ()) with - ()

| X ( ) |

( )

Digital Signal Processing

: is the magnitude spectrum

: is the phase spectrum

5

Frequency analysis of signals and systems

Determine and sketch the spectra of the following signal:

a) x(n) (n)

b) x(n) a nu(n) with |a|<1

X ( ) is periodic with period 2π.

X ( 2 k )

x ( n) e

j ( 2 k ) n

n

x(n)e jn X ( )

n

The frequency range for discrete-time signal is unique over the

frequency interval (-π, π), or equivalently, (0, 2π).

Remarks: Spectrum of discrete-time signals is continuous and

periodic.

Digital Signal Processing

6

Frequency analysis of signals and systems

Inverse discrete-time Fourier transform (IDTFT)

Given the frequency spectrum X ( ) , we can find the x(n) in timedomain as

1

x ( n)

2

X ( )e jn d

which is known as inverse-discrete-time Fourier transform (IDTFT)

Example: Consider the ideal lowpass filter with cutoff frequency wc.

Find the impulse response h(n) of the filter.

Digital Signal Processing

7

Frequency analysis of signals and systems

Properties of DTFT

Symmetry: if the signal x(n) is real, it easily follows that

X ( ) X ( )

or equivalently, | X () || X () |

(even symmetry)

(odd symmetry)

arg( X ()) arg( X ())

We conclude that the frequency range of real discrete-time signals can

be limited further to the range 0 ≤ ω≤π, or 0 ≤ f≤fs/2.

Energy density of spectrum: the energy relation between x(n) and

X(ω) is given by Parseval’s relation:

1

2

E x | x ( n) |

2

n

X ( ) d

2

S xx ( ) | X ( ) |2 is called the energy density spectrum of x(n)

Digital Signal Processing

8

Frequency analysis of signals and systems

Properties of DTFT

The relationship of DTFT and z-transform: if X(z) converges for

|z|=1, then

X ( z ) |z e x(n)e jn X ( )

j

n

Linearity: if

F

x1 (n)

X1 ( )

F

x2 (n)

X 2 ( )

F

a1 x1 (n) a2 x2 (n)

a1 X1 () a2 X 2 ()

then

Time-shifting: if

then

F

x(n)

X ( )

F

x(n k )

e jk X ( )

Digital Signal Processing

9

Frequency analysis of signals and systems

Properties of DTFT

F

X ( )

Time reversal: if x(n)

F

x(n)

X ( )

then

F

X1 ( )

Convolution theory: if x1 (n)

F

x2 (n)

X 2 ( )

then

F

x(n) x1 (n) x2 (n)

X () X1 () X 2 ()

Example: Using DTFT to calculate the convolution of the sequences

x(n)=[1 2 3] and h(n)=[1 0 1].

Digital Signal Processing

10

Frequency analysis of signals and systems

Frequency resolution and windowing

The duration of the data record is:

The rectangular window of length

L is defined as:

The windowing processing has two major effects: reduction in the

frequency resolution and frequency leakage.

Digital Signal Processing

11

Frequency analysis of signals and systems

Rectangular window

Digital Signal Processing

12

Frequency analysis of signals and systems

Impact of rectangular window

Consider a single analog complex sinusoid of frequency f1 and its

sample version:

With assumption

Digital Signal Processing

, we have

13

Frequency analysis of signals and systems

Double sinusoids

Frequency resolution:

Digital Signal Processing

14

Frequency analysis of signals and systems

Hamming window

Digital Signal Processing

15

Frequency analysis of signals and systems

Non-rectangular window

The standard technique for suppressing the sidelobes is to use a nonrectangular window, for example Hamming window.

The main tradeoff for using non-rectangular window is that its

mainlobe becomes wider and shorter, thus, reducing the frequency

resolution of the windowed spectrum.

The minimum resolvable frequency difference will be

where

window.

Digital Signal Processing

: c=1 for rectangular window and c=2 for Hamming

16

Frequency analysis of signals and systems

Example

The following analog signal consisting of three equal-strength

sinusoids at frequencies

where t (ms), is sampled at a rate of 10 kHz. We consider four data

records of L=10, 20, 40, and 100 samples. They corresponding of the

time duarations of 1, 2, 4, and 10 msec.

The minimum frequency separation is

Applying

the formulation

, the minimum length L to

resolve all three

sinusoids show be 20

samples for the rectangular window, and L =40 samples for the

Hamming case.

Digital Signal Processing

17

Frequency analysis of signals and systems

Example

Digital Signal Processing

18

Frequency analysis of signals and systems

Example

Digital Signal Processing

19

Frequency analysis of signals and systems

2. Discrete Fourier transform (DFT)

X ( ) is a continuous function of frequency and therefore, it is not a

computationally convenient representation of the sequence x(n).

DFT will present x(n) in a frequency-domain by samples of its

spectrum X ( ) .

A finite-duration sequence x(n) of length L has a Fourier transform:

L 1

X ( ) x(n)e jn

0 2

n 0

Sampling X(ω) at equally spaced frequency k 2 k , k=0, 1,…,N-1

where N ≥ L, we obtain N-point DFT of length N

L-signal:

L 1

2 k

X (k ) X (

) x(n)e j 2 kn / N

(N-point DFT)

N

n 0

DFT presents the discrete-frequency samples of spectra of discretetime signals.

Digital Signal Processing

20

Frequency analysis of signals and systems

2. Discrete Fourier transform (DFT)

With the assumption x(n)=0 for n ≥ L, we can write

N 1

X (k ) x(n)e j 2 kn / N , k 0,1,

, N 1.

(DFT)

n 0

The sequence x(n) can recover form the frequency samples by inverse

DFT (IDFT)

1 N 1

x(n) X (k )e j 2 kn / N , n 0,1,

N n 0

, N 1.

(IDFT)

Example: Calculate 4-DFT and plot the spectrum of x(n)=[1 1, 2, 1]

Digital Signal Processing

21

Frequency analysis of signals and systems

Matrix form of DFT

By defining an Nth root of unity WN e j 2 / N , we can rewritte DFT

and IDFT as follows

N 1

X (k ) x(n)WNkn , k 0,1,

, N 1.

(DFT)

n 0

1 N 1

x(n) X (k )WN kn , n 0,1,

N n 0

, N 1.

(IDFT)

Let us define: x(0)

X (0)

x(1)

X (1)

X

xN

N

x

(

N

1)

X

(

N

1)

The N-point DFT can be expressed in matrix form as: XN WN x N

Digital Signal Processing

22

Frequency analysis of signals and systems

Matrix form of DFT

1

1

1

1 W

2

W

N

N

WN 1 WN2

WN4

1 WNN 1 WN2( N 1)

WNN 1

WN2( N 1)

WN( N 1)( N 1)

1

Let us define: x(0)

X (0)

x(1)

X (1)

X

xN

N

x

(

N

1)

X

(

N

1)

The N-point DFT can be expressed in matrix form as: XN WN x N

Digital Signal Processing

23

Frequency analysis of signals and systems

Example: Determine the DFT of the four-point sequence x(n)=[1 1,

2 1] by using matrix form.

Digital Signal Processing

24

Frequency analysis of signals and systems

Properties of DFT

Properties

Time domain

Frequency domain

Notation

x ( n)

X (k )

Periodicity

Linearity

x(n N ) x(n)

X (k ) X (k N )

a1 x1 (n) a2 x2 (n)

a1 X1 (k ) a2 X 2 (k )

Circular time-shift

e j 2 kl / N X (k )

x((n l )) N

Circular convolution

Multiplication

of two sequences

Parveval’s theorem

Digital Signal Processing

1 N 1

Ex | x(n) | | X (k ) |2

N k 0

n 0

N

2

25

Frequency analysis of signals and systems

Frequency Analysis of Signals and Systems

Nguyen Thanh Tuan, Click

M.Eng.

to edit Master subtitle style

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: nttbk97@yahoo.com

Frequency analysis of signal involves the resolution of the signal into

its frequency (sinusoidal) components. The process of obtaining the

spectrum of a given signal using the basic mathematical tools is

known as frequency or spectral analysis.

The term spectrum is used when referring the frequency content of a

signal.

The process of determining the spectrum of a signal in practice base

on actual measurements of signal is called spectrum estimation.

The instruments of software programs

used to obtain spectral estimate of such

signals are kwon as spectrum analyzers.

Digital Signal Processing

2

Frequency analysis of signals and systems

The frequency analysis of signals and systems have three major uses

in DSP:

1) The numerical computation of frequency spectrum of a signal.

2) The efficient implementation of convolution by the fast Fourier

transform (FFT)

3) The coding of waves, such as speech or pictures, for efficient

transmission and storage.

Digital Signal Processing

3

Frequency analysis of signals and systems

Content

1. Discrete time Fourier transform DTFT

2. Discrete Fourier transform DFT

3. Fast Fourier transform FFT

Digital Signal Processing

4

Transfer functions

and Digital Filter Realizations

1. Discrete-time Fourier transform (DTFT)

The Fourier transform of the finite-energy discrete-time signal x(n) is

defined as:

X ( ) x(n)e jn

n

where ω=2πf/fs

The spectrum X(w) is in general a complex-valued function of

frequency:

X ( ) | X () | e j ( )

where () arg( X ()) with - ()

| X ( ) |

( )

Digital Signal Processing

: is the magnitude spectrum

: is the phase spectrum

5

Frequency analysis of signals and systems

Determine and sketch the spectra of the following signal:

a) x(n) (n)

b) x(n) a nu(n) with |a|<1

X ( ) is periodic with period 2π.

X ( 2 k )

x ( n) e

j ( 2 k ) n

n

x(n)e jn X ( )

n

The frequency range for discrete-time signal is unique over the

frequency interval (-π, π), or equivalently, (0, 2π).

Remarks: Spectrum of discrete-time signals is continuous and

periodic.

Digital Signal Processing

6

Frequency analysis of signals and systems

Inverse discrete-time Fourier transform (IDTFT)

Given the frequency spectrum X ( ) , we can find the x(n) in timedomain as

1

x ( n)

2

X ( )e jn d

which is known as inverse-discrete-time Fourier transform (IDTFT)

Example: Consider the ideal lowpass filter with cutoff frequency wc.

Find the impulse response h(n) of the filter.

Digital Signal Processing

7

Frequency analysis of signals and systems

Properties of DTFT

Symmetry: if the signal x(n) is real, it easily follows that

X ( ) X ( )

or equivalently, | X () || X () |

(even symmetry)

(odd symmetry)

arg( X ()) arg( X ())

We conclude that the frequency range of real discrete-time signals can

be limited further to the range 0 ≤ ω≤π, or 0 ≤ f≤fs/2.

Energy density of spectrum: the energy relation between x(n) and

X(ω) is given by Parseval’s relation:

1

2

E x | x ( n) |

2

n

X ( ) d

2

S xx ( ) | X ( ) |2 is called the energy density spectrum of x(n)

Digital Signal Processing

8

Frequency analysis of signals and systems

Properties of DTFT

The relationship of DTFT and z-transform: if X(z) converges for

|z|=1, then

X ( z ) |z e x(n)e jn X ( )

j

n

Linearity: if

F

x1 (n)

X1 ( )

F

x2 (n)

X 2 ( )

F

a1 x1 (n) a2 x2 (n)

a1 X1 () a2 X 2 ()

then

Time-shifting: if

then

F

x(n)

X ( )

F

x(n k )

e jk X ( )

Digital Signal Processing

9

Frequency analysis of signals and systems

Properties of DTFT

F

X ( )

Time reversal: if x(n)

F

x(n)

X ( )

then

F

X1 ( )

Convolution theory: if x1 (n)

F

x2 (n)

X 2 ( )

then

F

x(n) x1 (n) x2 (n)

X () X1 () X 2 ()

Example: Using DTFT to calculate the convolution of the sequences

x(n)=[1 2 3] and h(n)=[1 0 1].

Digital Signal Processing

10

Frequency analysis of signals and systems

Frequency resolution and windowing

The duration of the data record is:

The rectangular window of length

L is defined as:

The windowing processing has two major effects: reduction in the

frequency resolution and frequency leakage.

Digital Signal Processing

11

Frequency analysis of signals and systems

Rectangular window

Digital Signal Processing

12

Frequency analysis of signals and systems

Impact of rectangular window

Consider a single analog complex sinusoid of frequency f1 and its

sample version:

With assumption

Digital Signal Processing

, we have

13

Frequency analysis of signals and systems

Double sinusoids

Frequency resolution:

Digital Signal Processing

14

Frequency analysis of signals and systems

Hamming window

Digital Signal Processing

15

Frequency analysis of signals and systems

Non-rectangular window

The standard technique for suppressing the sidelobes is to use a nonrectangular window, for example Hamming window.

The main tradeoff for using non-rectangular window is that its

mainlobe becomes wider and shorter, thus, reducing the frequency

resolution of the windowed spectrum.

The minimum resolvable frequency difference will be

where

window.

Digital Signal Processing

: c=1 for rectangular window and c=2 for Hamming

16

Frequency analysis of signals and systems

Example

The following analog signal consisting of three equal-strength

sinusoids at frequencies

where t (ms), is sampled at a rate of 10 kHz. We consider four data

records of L=10, 20, 40, and 100 samples. They corresponding of the

time duarations of 1, 2, 4, and 10 msec.

The minimum frequency separation is

Applying

the formulation

, the minimum length L to

resolve all three

sinusoids show be 20

samples for the rectangular window, and L =40 samples for the

Hamming case.

Digital Signal Processing

17

Frequency analysis of signals and systems

Example

Digital Signal Processing

18

Frequency analysis of signals and systems

Example

Digital Signal Processing

19

Frequency analysis of signals and systems

2. Discrete Fourier transform (DFT)

X ( ) is a continuous function of frequency and therefore, it is not a

computationally convenient representation of the sequence x(n).

DFT will present x(n) in a frequency-domain by samples of its

spectrum X ( ) .

A finite-duration sequence x(n) of length L has a Fourier transform:

L 1

X ( ) x(n)e jn

0 2

n 0

Sampling X(ω) at equally spaced frequency k 2 k , k=0, 1,…,N-1

where N ≥ L, we obtain N-point DFT of length N

L-signal:

L 1

2 k

X (k ) X (

) x(n)e j 2 kn / N

(N-point DFT)

N

n 0

DFT presents the discrete-frequency samples of spectra of discretetime signals.

Digital Signal Processing

20

Frequency analysis of signals and systems

2. Discrete Fourier transform (DFT)

With the assumption x(n)=0 for n ≥ L, we can write

N 1

X (k ) x(n)e j 2 kn / N , k 0,1,

, N 1.

(DFT)

n 0

The sequence x(n) can recover form the frequency samples by inverse

DFT (IDFT)

1 N 1

x(n) X (k )e j 2 kn / N , n 0,1,

N n 0

, N 1.

(IDFT)

Example: Calculate 4-DFT and plot the spectrum of x(n)=[1 1, 2, 1]

Digital Signal Processing

21

Frequency analysis of signals and systems

Matrix form of DFT

By defining an Nth root of unity WN e j 2 / N , we can rewritte DFT

and IDFT as follows

N 1

X (k ) x(n)WNkn , k 0,1,

, N 1.

(DFT)

n 0

1 N 1

x(n) X (k )WN kn , n 0,1,

N n 0

, N 1.

(IDFT)

Let us define: x(0)

X (0)

x(1)

X (1)

X

xN

N

x

(

N

1)

X

(

N

1)

The N-point DFT can be expressed in matrix form as: XN WN x N

Digital Signal Processing

22

Frequency analysis of signals and systems

Matrix form of DFT

1

1

1

1 W

2

W

N

N

WN 1 WN2

WN4

1 WNN 1 WN2( N 1)

WNN 1

WN2( N 1)

WN( N 1)( N 1)

1

Let us define: x(0)

X (0)

x(1)

X (1)

X

xN

N

x

(

N

1)

X

(

N

1)

The N-point DFT can be expressed in matrix form as: XN WN x N

Digital Signal Processing

23

Frequency analysis of signals and systems

Example: Determine the DFT of the four-point sequence x(n)=[1 1,

2 1] by using matrix form.

Digital Signal Processing

24

Frequency analysis of signals and systems

Properties of DFT

Properties

Time domain

Frequency domain

Notation

x ( n)

X (k )

Periodicity

Linearity

x(n N ) x(n)

X (k ) X (k N )

a1 x1 (n) a2 x2 (n)

a1 X1 (k ) a2 X 2 (k )

Circular time-shift

e j 2 kl / N X (k )

x((n l )) N

Circular convolution

Multiplication

of two sequences

Parveval’s theorem

Digital Signal Processing

1 N 1

Ex | x(n) | | X (k ) |2

N k 0

n 0

N

2

25

Frequency analysis of signals and systems

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