Signal and Systems Prof. H. Sameti Chapter 5: The Discrete Time Fourier Transform Examples of the DT Fourier Transform Properties of the DT Fourier Transform

Signal and SystemsProf. H. Sameti

Chapter 5: The Discrete Time Fourier Transform• Examples of the DT Fourier Transform • Properties of the DT Fourier Transform • The Convolution Property and its Implication and

Uses• DTFT Properties and Examples• Duality in FS & FT

Book Chapter5: Section1

2

The Discrete-Time Fourier Transform

Computer Engineering Department, Signals and Systems

( 2 )Derivation:

[ ] Aperiodic and (for simplicity) of finite duration

is large enough so that

(Analogou

[ ] 0if 2

[ ] [ ]for 2and periodic wit

s to C

h

TFT except e )

•

•

•

j n j n

x n

N x

e

n n N

x n x n n N

period

[ ] [ ]for as

N

x n x n any n N


3

DTFT Derivation (Continued)


0

0

2

0 0

1

0

0 DTFT Synthesis Eq.

DTFT Analys

2[ ] ,

1[ ]

1 1[ ] [ ]

Define

( ) [ ] periodic in with period 2

1( )

is Eq.

jk nk

k N

jk nk

n N

Njk n jk n

n N n

j j n

n

jkk

x n a eN

a x n eN

x n e x n eN N

X e x n e

a X eN


4

DTFT Derivation (Home Stretch)


0 0 0 00

0 0

2

1 1[ ] ( ) ( ) ( )

2

As : [ ] [ ]for every

0,

The sum in ( ) an integral

The DTFT Pair

1[ ] ( ) Synthesis equation

2

( ) [ ]

k

jk jk n jk jk n

k N k N

a

j j n

j j n

x n X e e X e eN

N x n x n n

d

x n X e e d

X e x n e

Analysis equationn

Any 2π Interval in ω


5

DT Fourier Transform Pair


2

[ ] ( )

( ) [ ] Analysis Equation

FT

1[ ] ( ) Synthesis Equation

2 Inverse FT

j

j j n

n

j j n

x n X e

X e x n e

x n X e e d


6

Convergence Issues Synthesis Equation: None, since integrating over a finite

interval. Analysis Equation: Need conditions analogous to CTFT, e.g.


2[ ] Finite energy

or

[ ] Absolutely summable

n

n

x n

x n


7

ExamplesParallel with the CT examples in Lecture #8


0

0

0

0

1) [ ] [ ]

( ) [ ] 1

2) [ ] [ ] shifted unit sample

( )

Same amplitude (=1) as above, but with a phase - .

[ ]

j j n

n

j nj j n

n

x n n

X e n e

x n n n

X e n n

linear

e e

n


8

More Examples


0 01

2

2

Infinite sum formu

3) [ ] [ ], 1 Exponentially decaying function

X(e ) ( )

1 1=

1 (1 cos ) sin

1( )

1 2 cos1 1

0 : ( )11 2

1: ( )

la

1 2

j

n

j n j n j n

n nae

j

j

j

j

x n a u n a

a e ae

ae a ja

X ea a

X eaa a

X ea

2

1

1 aa


9

Still More


4) DT Rectangular pulse1(Drawn for 2)N

1 1

1 1

1( 2 )

1sin ( )

2( ) ( ) ( )sin( 2)

N Nj j n j n j

n N n N

NX e e e X e


10


1 sin[ ]

2

W j n

W

Wnx n e d

n

1

1[0] ( )

2

W j

W

Wx X e d

5)


11

DTFTs of Sums of Complex Exponentials


0

0

0

0

0

Recall CT result

What about DT:

a) We expect an impulse (of area 2 ) at

b) But ( ) must be periodic with period 2

In fact

: ( ) ( ) 2 ( )

[ ] ( ) ?

( ) 2 ( 2 )

j

j t

j n j

j

m

X e

x t e X j

x n e X e

X e m

00

( )

Note: The integration in the synthesis equation is over 2 period,

only need ( ) in 2 period. Thus,

1[ ] 2 ( 2 )

2j

j

j nj n

m

X e

X o ee n

x n m e d e


12

DTFT of Periodic Signals


[ ] [ ]x n x n N

0

0

0

0

0

0

DTFS Synthesis Eq.2

[ ] ,

From the last page: 2 ( 2 )

( ) [2 ( 2 )]

22 ( ) 2 ( )

jk nk

k N

jk n

m

jk

k N m

k kk k

x n a eN

e k m

X e a k m

ka k a

N

Linearity of DTFT


13

Example #1: DT sine function

Computer Engineering Department, Signal and Systems

𝑥 [𝑛]=sin𝜔0𝑛=1

2 𝑗𝑒 𝑗𝜔 0𝑛−

12 𝑗𝑒− 𝑗 𝜔0𝑛

𝑋 (𝑒 𝑗 𝜔)= 𝜋𝑗 ∑𝑚=−∞

∞

𝛿 (𝜔−𝜔0 −2𝜋𝑚 )− 𝜋𝑗 ∑𝑚=− ∞

∞

𝛿 (𝜔+𝜔0 −2𝜋𝑚)


14

Example #2: DT periodic impulse train


𝑥 [𝑛 ]= ∑𝑘=−∞

∞

𝛿 [𝑛−𝑘𝑁 ] 𝜔0=2𝜋𝑁

𝑎𝑘=1𝑁 ∑

𝑛=⟨𝑁 ⟩𝑥 [𝑛]𝑒− 𝑗𝑘𝜔 0𝑛

¿ 1𝑁 ∑

𝑛=0

𝑁− 1

𝑥[𝑛] 𝑒− 𝑗𝑘𝜔0𝑛= 1𝑁

¿𝑋 (𝑒 𝑗 𝜔)=2𝜋

𝑁 ∑𝑘=− ∞

∞

𝛿(𝜔−2𝜋𝑘𝑁 )

⇓

— Also periodic impulse train – in the frequency domain!


15

Properties of the DT Fourier Transform


𝑋 (𝑒 𝑗 𝜔)= ∑𝑛=− ∞

∞

𝑥 [𝑛]𝑒− 𝑗 𝜔𝑛

𝑥 [𝑛 ]=1

2𝜋 2𝜋

𝑋 (𝑒 𝑗 𝜔)𝑒 𝑗 𝜔𝑛𝑑𝜔

— Analysis equation

— Synthesis equation

1) Periodicity:𝑋 (𝑒 𝑗 (𝜔+2𝜋 ) )=𝑋 (𝑒 𝑗 𝜔 ) — Different from CTFT

2) Linearity:𝑎𝑥1[𝑛]+𝑏𝑥2[𝑛 ]↔𝑎 𝑋1 (𝑒 𝑗 𝜔 )+𝑏𝑋 2 (𝑒 𝑗 𝜔 )


16

More Properties

3) Time Shifting: 4) Frequency Shifting: - Important implications in DT because of periodicity Example


𝑥 [𝑛−𝑛0] ↔𝑒− 𝑗𝜔 𝑛0𝑋 (𝑒 𝑗 𝜔 )

𝑒 𝑗 𝜔0𝑛𝑥 [𝑛]↔ 𝑋 (𝑒 𝑗 (𝜔−𝜔0 ) )


17

Still More Properties

5) Time Reversal:

6) Conjugate Symmetry:


𝑥 [−𝑛]↔𝑋 (𝑒− 𝑗 𝜔 )

𝑥 [𝑛 ] 𝑟𝑒𝑎𝑙 ⇒ 𝑋 (𝑒 𝑗𝜔 ) = 𝑋∗ (𝑒− 𝑗𝜔 )

¿ 𝑋 (𝑒 𝑗𝜔 )∨ 𝑎𝑛𝑑 ℜ {𝑋 (𝑒 𝑗𝜔 )} 𝑎re 𝑒𝑣𝑒𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠

∠ 𝑋 (𝑒 𝑗𝜔 ) 𝑎𝑛𝑑 ℑ {𝑋 (𝑒 𝑗 𝜔 )} 𝑎𝑟𝑒 𝑜𝑑𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠

𝑥 [𝑛 ] 𝑟𝑒𝑎𝑙 𝑎𝑛𝑑 𝑒𝑣𝑒𝑛 ⇔ 𝑋 (𝑒 𝑗𝜔 ) 𝑟𝑒𝑎𝑙 𝑎𝑛𝑑 𝑒𝑣𝑒𝑛𝑥 [𝑛 ] 𝑟𝑒𝑎𝑙 𝑎𝑛𝑑 𝑜𝑑𝑑 ⇔ 𝑋 (𝑒 𝑗 𝜔 ) 𝑝𝑢𝑟𝑒𝑙𝑦 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑎𝑛𝑑 𝑜𝑑𝑑

and


18

Yet Still More Properties


7) Time Expansion Recall CT property:

𝑥 (𝑎𝑡 ) ↔ 1

¿ 𝑎∨¿ 𝑋 ( 𝑗 (𝜔𝑎 ))¿Time scale in CT is infinitely fine

But in DT: x[n/2] makes no sense x[2n] misses odd values of x[n]

But we can “slow” a DT signal down by inserting zeros:k —an integer ≥ 1x(k)[n] — insert (k - 1) zeros between successive values

Insert two zeros in this example

(k=3)


19

Time Expansion (continued)


𝑥(𝑘) [𝑛 ]={𝑥 [𝑛𝑘 ]𝑖𝑓 𝑛𝑖𝑠 𝑎𝑛𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑘0𝑜𝑡 h𝑒𝑟𝑤𝑖𝑠𝑒 — Stretched by a

factor of k in time domain

𝑋 (𝑘 ) (𝑒𝑗𝜔 ) = ∑

𝑛=− ∞

∞

𝑥 (𝑘 ) [𝑛]𝑒− 𝑗 𝜔𝑛 ´́

𝑛=𝑚𝑘 ∑𝑚=−∞

∞

𝑥 (𝑘) [𝑚𝑘]𝑒− 𝑗 𝜔𝑚𝑘

¿ ∑𝑚=−∞

∞

𝑥[𝑚]𝑒− 𝑗 (𝑘𝜔 )𝑚=𝑋 (𝑒 𝑗𝑘𝜔 ) -compressed by a factor of k in frequency domain


20

Is There No End to These Properties?


8) Differentiation in Frequency

𝑋 (𝑒 𝑗𝜔 ) = ∑𝑛=− ∞

∞

𝑥 [𝑛]𝑒− 𝑗 𝜔𝑛

𝑑𝑑𝜔

𝑋 (𝑒 𝑗 𝜔 ) = − 𝑗 ∑𝑛=− ∞

∞

𝑛𝑥 [𝑛 ]𝑒− 𝑗𝜔𝑛

⇓multiply by j on both sides

Multiplication by n

𝑛𝑥 [𝑛] ↔ 𝑗𝑑𝑑𝜔

𝑋 (𝑒 𝑗 𝜔 ) DifferentiationIn frequency

9) Perseval’s Relation

∑𝑛=− ∞

∞

¿𝑥 [𝑛]¿2 = 12𝜋

2 𝜋

¿ 𝑋 (𝑒 𝑗𝜔 )¿2𝑑𝜔

Total energy in time domain Total energy in frequency domain


21

The Convolution Property


¿h[n]

𝑌 (𝑒 𝑗 𝜔)=𝐻 (𝑒 𝑗 𝜔)𝑋 (𝑒 𝑗 𝜔 )

⇒ 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝐻 (𝑒 𝑗 𝜔 ) = 𝐷𝑇𝐹𝑇 𝑜𝑓 𝑡 h𝑒 𝑢𝑛𝑖𝑡 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒Example #1: 𝑥 [𝑛 ]=𝑒 𝑗 𝜔0𝑛↔ 𝑋 (𝑒 𝑗 𝜔 ) = 2𝜋 ∑

𝑘=−∞

∞

𝛿 (𝜔−𝜔0 −2𝜋𝑘 )

𝑌 (𝑒 𝑗𝜔 ) = 𝐻 (𝑒 𝑗𝜔 ) 2𝜋 ∑𝑘=−∞

∞

𝛿 (𝜔−𝜔0 −2𝜋𝑘 )

¿ 2𝜋 ∑𝑘=− ∞

∞

𝐻 (𝑒 𝑗 (𝜔0+2𝜋𝑘 ) )𝛿 (𝜔−𝜔0 − 2𝜋 𝑘)

´́𝐻 𝑃𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝐻 (𝑒 𝑗𝜔0 ) 2𝜋 ∑

𝑘=−∞

∞

𝛿 (𝜔−𝜔0 −2𝜋𝑘 )

⇓𝑦 [𝑛] = 𝐻 (𝑒 𝑗𝜔 0 )𝑒 𝑗𝜔0𝑛


22

Example#2: Ideal Lowpass Filter


h [𝑛 ]=1

2𝜋 −𝜔 𝑐

𝜔𝑐

𝑒 𝑗 𝜔𝑛𝑑𝜔=sin𝜔𝑐𝑛𝜋𝑛


23Computer Engineering Department, Signal and Systems

Example #3:

sin( 𝜋𝑛4 )𝜋 𝑛

∗sin ( 𝜋𝑛2 )𝜋𝑛

=sin ( 𝜋𝑛4 )𝜋𝑛

⇓ ⇑

Book Chapter#: Section#

24

Convolution Property Example


Ratio of polynomials in

A, B determined by partial fraction expansion (PFE)

[ ] [ ]nh n u n [ ] [ ]nx n u n | |,| | 1

1( )

1j

jH e

e

1( )

1j

jX e

e

1 1[ ] [ ]* [ ] ( ) ( )( )

1 1j

j jy n h n x n Y e

e e

je

: ( )1 1

jj j

A BY e

e e

[ ] [ ] [ ]n ny n A u n B u n


25


21: ( ) ( )

1

( )[ ] [ ]

[ ] ( 1) [ ]

jj

j

n

Y ee

dX ey n nx n j

d

y n n u n


26

DT LTI system Described by LCCDE’s

From time-shifting property:


0 0

[ ] [ ]N M

k kk k

a y n k b x n k

[ ] ( )j k jx n k e X e

0 0

( ) ( )N M

j k j j k jk k

k k

a e Y e b e X e

0

0

( ) ( )

Mj k

kj jk

Nj k

kk

b eY e X e

a e

( )jH e

Rational function of use PFE to get h[n]

je


27

Example: First-order recursive system

With the condition of initial rest causal


[ ] [ 1] [ ]y n y n x n | | 1

(1 ) ( ) ( )j j je Y e X e

( ) 1( )

( ) 1

jj

j j

Y eH e

X e e

[ ] [ ]nh n u n


28

DTFT Multiplication Property

Derivation:


( )1 2 1 2

2

1[ ] [ ]. [ ] ( ) ( ) ( )

2j j jy n x n x n Y e X e X e d

1 2

1( ) ( )

2j jX e X e

Periodic Convolution

1 2( ) [ ]. [ ]j j n

n

Y e x n x n e

1 2

2

1( ( ) ) [ ]2

j j n j n

n

X e e d x n e

( )

1 2

2

1( ( ) [ ] )

2j j n

n

X e x n e d

( )

2 ( )jX e

( )1 2

2

1( ) ( )

2j jX e X e d


29

Calculating Periodic Convolutions

Suppose we integrate from –π to π:

where


11

( ) | |ˆ ( )0

jj X e

X eotherwise

( )1 2

( )1 2

1( ) ( ) ( )

2

1 ˆ ( ) ( )2

j j j

j j

Y e X e X e d

X e X e d


30

Example:


21 2

1 2

1 2

sin( )4[ ] ( ) [ ]. [ ]

sin( )4[ ] [ ]

1( ) ( ) ( )

2j j j

ny n x n x n

nn

x n x nn

Y e X e X e


31

Duality in Fourier Analysis Fourier Transform is highly symmetric CTFT: Both time and frequency are continuous and in general aperiodic

Suppose f (.) and g (.) are two functions related by:

Then: Let т = t and r = ω: Let т = -ω and r = t:


Same except for these differences

1 1( ) ( ) ( ) ( )x t g t X j f

( ) ( ) jrf r g e d

2 2( ) ( ) ( ) 2 ( )x t f t X j g

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt


32

Example of CTFT duality



33

DTFS Discrete & periodic in time Periodic & discrete in frequency

Duality in DTFS Suppose f(.) and g(.) are two functions related by:

Then:

Let m = n and r = -k: Let r = n and m = k:


00

2[ ] [ ],jk n

kk N

x n a e x n NN

0

1[ ] jk n

k k Nk N

a x n e aN

0 01

[ ] [ ] [ ] [ ]j r m j r m

r N m N

f m g r e g r f m eN

1

1[ ] [ ] [ ]kx n f n a g k

N

2[ ] [ ] [ ]kx n g n a f k


34

Duality between CTFS and DTFT

CTFS:

DTFT:


00

2( ) [ ],

kjk t

kk

x t a e x t TT

01

( ) jk tk

T

a x t e dtT

Periodic in time Discrete in frequency

2

1[ ] ( )

2j j nx n X e e d

( 2 )( ) [ ] ( )j j n j

n

X e x n e X e

Discrete in time Periodic in frequency


35

CTFS-DTFT Duality

Suppose f(.) is a CT signal and g[.] a DT sequence related by :

Then:


( ) [ ] ( 2 )j

m

f g m e f

( ) ( ) [ ]kx t f t a g k

periodic with period 2π

[ ] [ ] ( ) ( )jx n g n X e f

Documents

Signal and Systems Prof. H. Sameti Chapter 5: The Discrete Time Fourier Transform Examples of the DT Fourier Transform Properties of the DT Fourier Transform