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
Book Chapter5: Section1
3
DTFT Derivation (Continued)
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
4
DTFT Derivation (Home Stretch)
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
5
DT Fourier Transform Pair
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
6
Convergence Issues Synthesis Equation: None, since integrating over a finite
interval. Analysis Equation: Need conditions analogous to CTFT, e.g.
Computer Engineering Department, Signals and Systems
2[ ] Finite energy
or
[ ] Absolutely summable
n
n
x n
x n
Book Chapter5: Section1
7
ExamplesParallel with the CT examples in Lecture #8
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
8
More Examples
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
9
Still More
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
10
Computer Engineering Department, Signals and Systems
1 sin[ ]
2
W j n
W
Wnx n e d
n
1
1[0] ( )
2
W j
W
Wx X e d
5)
Book Chapter5: Section1
11
DTFTs of Sums of Complex Exponentials
Computer Engineering Department, Signals and Systems
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
Book Chapter5: Section1
12
DTFT of Periodic Signals
Computer Engineering Department, Signals and Systems
[ ] [ ]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
Book Chapter5: Section1
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)
Book Chapter5: Section1
14
Example #2: DT periodic impulse train
Computer Engineering Department, Signal and Systems
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!
Book Chapter5: Section1
15
Properties of the DT Fourier Transform
Computer Engineering Department, Signal and Systems
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 )
Book Chapter5: Section1
16
More Properties
3) Time Shifting: 4) Frequency Shifting: - Important implications in DT because of periodicity Example
Computer Engineering Department, Signal and Systems
𝑥 [𝑛−𝑛0] ↔𝑒− 𝑗𝜔 𝑛0𝑋 (𝑒 𝑗 𝜔 )
𝑒 𝑗 𝜔0𝑛𝑥 [𝑛]↔ 𝑋 (𝑒 𝑗 (𝜔−𝜔0 ) )
Book Chapter5: Section1
17
Still More Properties
5) Time Reversal:
6) Conjugate Symmetry:
Computer Engineering Department, Signal and Systems
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
Book Chapter5: Section1
18
Yet Still More Properties
Computer Engineering Department, Signal and Systems
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)
Book Chapter5: Section1
19
Time Expansion (continued)
Computer Engineering Department, Signal and Systems
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
Book Chapter5: Section1
20
Is There No End to These Properties?
Computer Engineering Department, Signal and Systems
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
Book Chapter5: Section1
21
The Convolution Property
Computer Engineering Department, Signal and Systems
¿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
Book Chapter5: Section1
22
Example#2: Ideal Lowpass Filter
Computer Engineering Department, Signal and Systems
h [𝑛 ]=1
2𝜋 −𝜔 𝑐
𝜔𝑐
𝑒 𝑗 𝜔𝑛𝑑𝜔=sin𝜔𝑐𝑛𝜋𝑛
Book Chapter5: Section1
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
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
25
Computer Engineering Department, Signal and Systems
21: ( ) ( )
1
( )[ ] [ ]
[ ] ( 1) [ ]
jj
j
n
Y ee
dX ey n nx n j
d
y n n u n
Book Chapter#: Section#
26
DT LTI system Described by LCCDE’s
From time-shifting property:
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
27
Example: First-order recursive system
With the condition of initial rest causal
Computer Engineering Department, Signal and Systems
[ ] [ 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
Book Chapter#: Section#
28
DTFT Multiplication Property
Derivation:
Computer Engineering Department, Signal and Systems
( )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
Book Chapter#: Section#
29
Calculating Periodic Convolutions
Suppose we integrate from –π to π:
where
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
30
Example:
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
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:
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
32
Example of CTFT duality
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
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:
Computer Engineering Department, Signal and Systems
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
Book Chapter#: Section#
34
Duality between CTFS and DTFT
CTFS:
DTFT:
Computer Engineering Department, Signal and Systems
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