presentation-6 freq. domain...

MEH329DIGITAL SIGNAL PROCESSING

-6-Frequency Domain Representation of DT

Signals and Systems - 1

Frequency Domain Representation

• Sinusoidals and complex exponentials play keyrole in representing DT signals.

• Complex exponentials are eigenfunctions of LTI systems.

MEH329 Digital Signal Processing 2

LTI System j nx n e j j ny n H e e


• Convolution sum:


k k

y n x k h n k h k x n k

j nx n e j n k

k

y n h k e

j n j k

k

y n e h k e

j n je H e

Eigen func. Eigen value


• Frequency response:


j j k

k

H e h k e

j

j j jR I

j H ej

H e H e jH e

H e e


• For a real system:


*

*

j j

j j k j k

k k

j k j

k

j H e j H ej j

j j

j j

H e h k e h k e

h k e H e

H e e H e e

H e H e

H e H e


• Example: Ideal delay system


dy n x n n

d

d

j n n

j nj n

y n e

e e

dj njH e e

• Alternatively;

dj nj j k j kd

k k

H e h k e k n e e


• From the Euler relation:


cos

sin

jR d

jI d

H e n

H e n

• The magnitude and phase:

1j

jd

H e

H e n

linear


• From the principle of superposition, broadclass of signals can be represented as linearcombinations of complex exponentials:


kj nk

k

x n a e

• Output of an LTI system:

k kj j nk

k

y n a H e e


• Example: Ideal delay system with the input:


0cosx n A n

0 0

2 2j n j nj jA A

x n e e e e

0

0

1 2

1

2

2

2

j nj

j nj

x n x n x n

Ax n e e

Ax n e e



0 0

0 0

1

2

2

2

j j nj

j j nj

Ay n H e e e

Ay n H e e e

0 0 0 0

1 2

2 2j j n j j nj j

y n y n y n

A AH e e e H e e e

0 00cosj jA H e n H e


• For the ideal delay system:


dj njH e e

0 00

0 0

0

cos

cos

cos

j j

d

d

y n A H e n H e

A n n

A n n

0

0

dj njH e e


• Example: Moving average system:


2

11 2

1

1

Mj j k

k M

H e eM M


• Using geometric series expansion:


2 1

1 2 1 2

2 1

1 2 1 2

2 1

2 1

1

1 2

1 /2 1 /21 /2

1 2

1 /2 1 /2/2

/2 /21 2

1 2 /2

1 2

1

1 1

1

1 1

1

1

sin 1 / 21

1 sin / 2

j M j Mj

j

j M M j M Mj M M

j

j M M j M Mj M M

j j

j M M

e eH e

M M e

e ee

M M e

e ee

M M e e

M Me

M M


• Magnitude and Phase responses for the caseM1=0 and M2=4:


jH e

jH e


• For an arbitrary system:


• Since the frequency response of a discrete-time LTI system is always periodic, we need to only consider a period of length 2π• We will consider the interval −π < Ω ≤ , in which case low frequencies are close to zero, and high frequencies are close to ±


Frequency Domain RepresentationFilter Types

• We need constant magnitude and linear phasefor the frequencies of interest:


Ω

0( )jH e

Ω

0( )jH e

LPF

Ω

0( )jH e

HPF

Ω

0( )jH e

BPF

Ω

0( )jH e

BSF

Frequency Domain RepresentationSuddenly Applied Complex Exponential Inputs

• If


j nx n e u n

k

y n h k x n k

0

0

0 1

nj n k

k

nj n j k

k

j n j k j k

k k n

y n h k e

e h k e

e h k e h k e

• If we consider the output for n≥0

Frequency Domain RepresentationSuddenly Applied Complex Exponential Inputs


1

j j n j n j k

k n

ss t

y n H e e e h k e

y n y n

Steady-state responsewhen the input is equal to Ω for all n.

Transient responseIf the impulse respones has finite length, the transient response approaches zero

• If the impulse response finite length (0 ≤ n≤M): , 1j j n

ssy n y n H e e n M

Frequency Domain RepresentationFourier Transform

Jean Baptiste Joseph Fourier was born in France in 1768. He attended the Ecole Royale Militaire and in 1790 became a teacher there. Fourier continued his studies at the EcoleNormale in Paris, having as his teachers Lagrange, Laplace, and Monge. Later on, he, together with Monge and Malus, joined Napoleon as scientific advisors to his expedition to Egypt where Fourier established the Cairo Institute.

In 1822 Fourier has published his most famous work: The Analytical Theory of Heat. Fourier showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series.


Frequency Domain RepresentationFourier Transform

• FT pair in continuous time:


j tX j x t e dt

1

2j tx t X j e d

Frequency Domain RepresentationDiscrete Time Fourier Transform

Fourier spectrum/Freq. Spectrum/Spectrum


cos sin

j j n

n

n

X e x n e

x n n j n

• Discrete Time Fourier Transform:



• Periodicity:

2 2

2

[ ]

[ ] [ ]

j r j r n

n

j n j rn j n

n n

j

X e x n e

x n e e x n e

X e



j

j j jR I

j X ej

X e X e jX e

X e e

deeXnx njj

2

1][

• Inverse Transform:



• Example: x n n

0

[ ]

[0]

1

j j n

n

j

X e n e

e



• Example: 3 2 2 1 3x n n n n n

2 0 1 3

2 3

3 2 2 1 3

3 2

3 1 2

j j n

n

j j j j

j j j

X e n n n n e

e e e e

e e e



• Example: 1 , 0

0 , otherwise

n Nh n

1

0

1

1

1

1

Nj j n

n

j N

j

j N

j

X e e

e

e

e

e



• Example: nx n a u n

0 0

1

0

1lim lim

1

nj n j n j

n n

NjN njjN N

n

X e a e ae

aeae

ae

11 1 , 1

1j j j

jae e X e a

ae



• Convergence:

1 i.e.

converges absolutely or uniformly

n

j

if x n l x n

X e

.

( lim 0)k

j j nk

n

j jkk

ak

bs error

Let X eX e x Xn e e



• Since:( )jX e

j j n j n

n n

X e x n e x n e

j

n

X e x n

• This is a sufficient condition for the existence of DTFT.• FIR systems always have FT, but this is not true for IIR

systems.



• Absolutely summable sequences always havefinite energy.

• However, finite energy sequences are notnecessary absolutely summable.

2

1 2

0

. , :

cos( )

n

n

u n

a arenot l In fact they arenot l x n

A n

must be



• Convergence in terms of means square error(MSE):

• The total energy of the error must approachzero, not an error itself!

• In this case, we say that

2

lim 0j jkk

energy of the error

X e X e

jFT x n X e



• Example: 11x n u n

n

?

1

1 diverges!

n n

x nn

?2

2

21

1 converges (finite energy)!

6

n

n

x n

n

(Harmonic series)

(p - series)



• Example: Ideal low-pass filterΩ

0( )jH e

20

1

Ωc

1,

0,

cjLP

c

H e

1 1

2 2

1

2

sin,

c c

c c

c c

j j nLP

j n j n

c

h n H e d e d

e ejn

nn

n

• The filter is not causal and have infinite length(cannot be implemented practically!)



• Reversely:

sinj j ncLP

n

nH e e

n

• For the practical implementation, we cantruncate the sample numbers:

sinLP

MM j j nc

n M

nH e e

n



• Example:• Not absolutely summable• Not even square summable

1x n

2 2j

r

X e r

• Let’s place into inverse DTFT equation

0

1

2

1 2 2

2

1

j j n

j n

r

j n j n

x n X e e d

r e d

e d e

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