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Fourier Analysis of Discrete-Time Signals
• Discrete-time signals processing
• Fourier analysis
• Fast Fourier Transform algorithm
1-Fourier series for discrete-time periodic signals2-Discrete-Time Fourier Transform for aperiodic signals
2
The response of discrete-time linearinvariant systems to the complex
exponential with unitary magnitude
( ) [ ] j k
kH h k e
∞− Ω
=−∞
Ω = ⋅∑
H(Ω)0j ne Ω ( )00
j ne HΩ ⋅ Ω
Eigenfunction for any discrete time LTI
Eigenvalue
2
3
[ ] [ ] [ ] ( ) [ ] .0000 ∑ ∑∞
−∞=
∞
−∞=
Ω−Ω−ΩΩ ⋅⋅=⋅=∗=k k
kjnjknjnj ekheekhenhny
( ) [ ]Notation: j k
kH h k e
∞− Ω
=−∞
Ω = ⋅∑
[ ] ( ) ( ) ( )[ ]000 00ΩΦ+ΩΩ ⋅Ω=Ω⋅= njnj eHHeny
• proof
4
linear combination of discrete-time complex exponentials with unitary magnitudes, different frequencies
⇒ response of the discrete-time LTI system - linear combination of partial responses, H(Ωk) Φk[n]:
H(Ω)kj n
kk
a e Ω∑ ( ) kj nk k
ka H e ΩΩ∑
3
5
Fourier series for discrete-timeperiodic signals
x[n] is periodic of period N if
x[n+N]=x[n], for any integer n.
In a period N values x[0], x[1], x[N-1].
x[0]= x[N], x[N+1]= x[1] ...
x[n]= x[(n)N]
(n)N - n in the class of quotient (remainder) modulo N. For n<0, the quotient (n)N positive.
6
The space of periodic discrete-time signals of period N is N dimensional => N-dimensional basis. Orthogonal basis
[ ]⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−≤≤ℵ∈=Φπ
10 , 2
Nkkenn
Njk
k
i.e. it’s orthogonal and complete:
unique decomposition ck.
Nπ
=Ω2
0
[ ] [ ] ,,
0,k l
N k ln n
k l=⎧
Φ Φ = ⎨ ≠⎩
[ ] ,1
00∑
−
=
Ω=N
k
njkkecnx
4
7
Proof
• Orthogonality
[ ] [ ] ( )( )( )( )
( )( ) 00
0
000
11
11
,
2
1
0
1
0
Ω−
π−≠
Ω−
Ω−
−
=
Ω−Ω−−
=
Ω
−
−=
−
−=
==⋅=ΦΦ ∑∑
lkj
lkjkl
lkj
Nlkj
N
n
nlkjnjlN
n
njklk
ee
ee
eeenn
8
[ ] [ ]( )( ) .0
11,
0
2=
−
−=ΦΦ
Ω−
π−≠
lkj
lkjkllk
eenn
For k=l:
[ ] [ ] .1,1
0
1
000 Neenn
N
n
N
n
njknjkkk ==⋅=ΦΦ ∑∑
−
=
−
=
Ω−Ω
For k≠l :
[ ] [ ] [ ] [ ]. 1,0in and ,0,
, 22
2−=
⎩⎨⎧
Φ≠=
=ΦΦ NlNnlklkN
nn klk
5
9
Completeness.
Any discrete-time periodic signal has the form:
[ ] ,1
00∑
−
=
Ω=N
k
njkkecnx
with unique coefficients ck. With the notations: njkk
nnj
n ee 00 and ΩΩ =φ=φthe last relation becomes for n=0, n=1,…,n=N-1
-linear system of
-N equations and
-N unknowns c0 , c1,…, cN-1.
10
The determinant of the eq.:
Representing the N discrete-time complex exponential with unitary magnitude in the complex plane, the following figure is obtained.
6
11
It can be seen that:( )
solution. unique has equations of system theand 0, zero,not ist determinan theso ,0 for , ≠Δ≠φ−φ≠∀ lklklk
So, the considered set of complex exponentials is also complete.Being orthogonal and complete it is a basis.
12
The decomposition of the periodic discrete-time signal x[n] in the considered basis = decomposition in Fourier series:
[ ] [ ]
[ ][ ] ;1, 1
02
2
0∑−
=
Ω−⋅=φ
φ=
N
n
njk
k
kk enx
Nn
nnxc
The sequence of coefficients is also periodic with same period.
[ ] kcnx ↔
, 0 1.k N kc c k N+ = ≤ ≤ −
!!!definition
7
13
• proof
[ ]( )
[ ]
[ ] .10 ,1 and 1But
11
1
0
22
1
0
221
0
2
−≤≤=⋅==
⋅⋅=⋅=
∑
∑∑
−
=
π−
+π−
−
=
π−π
−−
=
π+−
+
NkcenxN
ce
eenxN
enxN
c
kN
n
nN
jkNk
nj
N
n
njnN
jkN
n
nN
NkjNk
14
Examples
[ ] ⎟⎠⎞
⎜⎝⎛ π
= nN
nx 2sin
( ).
21
21
21
212sin
21222 nN
NjnN
jnN
jnN
je
je
je
je
jn
N
π−−
ππ−
π
−=−=⎟⎠⎞
⎜⎝⎛ π
1. The signal
Euler’s relation :
Identification :
1 1
0 2 2
1 1, ; 2 2
... 0.
N
N
c cj j
c c c
−
−
= = −
= = = =
8
15
Spectral diagrams, N=6
16
Euler’s relation :
identification:
9
17
3. Periodic signal, period N.
definition:
For k=0:
18
the sum of a geometric series
10
19
•truncation of the Fourier series of a discrete-time signal, an approximation of this signal.
•better for higher number of terms.
•all N terms no error.
•example 3 , N = 9 and 2N1+1 = 5 . Fourier coefficients : c0 = 0,556 ,
c1 = c8 =0,32 ,
c2 = c7 = -0,059 ,
c3 = c6 = -0,111, and
c4 = c5 = 0,073.
20
The truncated signal with 2M+1 terms from its Fourier series:
For M=1, 2, 3 and 4 :
11
21
22
Properties of Discrete-Time Fourier Series
1. Linearity
x[n] and y[n] periodic with same period
Fourier coefficients ckx and ck
y
ax[n]+by[n] - same period.
Homework: Prove it.
12
23
2. Time Shifting
Dual operations: time shifting ↔ modulation in frequency (multiplication with a complex exponential).
The absolute value of the Fourier coefficients is not affected by time shifting.
24
Dual operations: complex conjugation in time ↔ reversal and complex conjugation in frequency domain.
3. Complex conjugation
13
25
Dual operations: time reversal ↔ reversal in frequency.
Reversal is auto-dual.
4. Time Reversal
26
5. Time scale modification
Different than in continuous-time domain (the scale factor can’t be any real number).
One way :
signal x[n] periodic - period N, => signal x(m)[n] periodic N’=mN.
Proof.
14
27
time dilation / interpolation : insert m-1 zeros between two consecutive samples of x[n]
The period of x(m)[n] is N’=mN, m times higher than the period of the signal x [n]
The fundamental frequency of the new signal - m times smaller.
Dual operations : discrete-time dilation ↔ frequency compression with the same scale factor.
28
The second way for the scale modification of a discrete-time signal is a time compression with information loss :
[ ] [ ]mnxnx m =)(
15
29
6. Signal’s modulation
dual operation of shifting.
product of signal with the complex exponential = signal modulation.
30
7. Signals’ product
circular (or periodic) convolution of discrete-time periodic sequences, see nextslide.
16
31
8. Circular convolution of signals
- dual operation of multiplicationx[n] and y[n] periodic withperiod N, Fourier coefficients ck
x and cky
The signal z[n] is also periodic with period N.
32
9. Discrete-time Differentiation
defined by the first order difference of the signal x[n] -difference between signal and its delayed version with 1.
If the original signal is periodic than its first order difference is also periodic with same period.
Homework: Prove it.
17
33
10. Addition in time domain
The sum of samples of x [n] with period N , with no DC component ( ) has the same period :
Proof.
sum of the samples over one period
[ ] [ ] [ ] [ ]∑ ∑ ∑+
−∞= −∞=
+
+=+==+
Nn
m
n
m
Nn
nmmxmxmxNny
1
[ ] ;1 1
00∑
−
=
Ω−⋅=N
n
njkxk enx
Nc
For k=0:[ ] [ ]
1
00 1
1 0 0N n N
x
n m nc x n x m
N
− +
= = +
= = ⇒ =∑ ∑
34
discrete-time differentiation of y[n], y[n]-y[n-1]= x[n]:
[ ] [ ]∑−
−∞==−
11
n
mmxny
For k=0 :
18
35
11. Properties of the Fourier coefficients of the real signals
If x[n] is real then it equals its complex conjugate. So its Fourier coefficients will equal the Fourier coefficients of its complex conjugate given by the property number 3.
Polar form:
Cartesian form:
The even and odd parts of a real signal:
36
Parseval’s Relation
the square of the l2 norm of the considered signal.
-the power of the periodic signal - can be computed in the time domain or in the frequency domain.
19
37
Discrete-Time Fourier Transform
non-periodic signal → periodic signal by repetition:
N →∞, periodic signal → non-periodic one.
[ ] [ ]k
x n x n kN∞
=−∞= −∑
38
Definition
Discrete-Time Fourier Transform.
Inverse Discrete-Time Fourier Transform.
20
39
The Fourier expansion of a periodic signal :
its Fourier coefficients are :
for the N values of k belonging to an interval which length is Nthe periodic signal and the original one are equals. For the rest of the values of n the original signal equals zero:
40
notation:
the corresponding periodic signal :
At the limit N →∞ (Ω0 → 0) :
21
41
The periodic signal is:
where:
For the signal considered :
the function X(Ω) :
42
The spectrum of the original signal is the envelope X(Ω).
22
43
Discrete-Time Fourier Transform.
If the signal x[n] belongs to l1 then the DTFT is convergent.
Proof.
( ) [ ] [ ] [ ] [ ] 1nxnxenxenxXnn
jn
n
jn ==⋅≤⋅=Ω ∑∑∑∞
−∞=
∞
−∞=
Ω−∞
−∞=
Ω−
The values of the DTFT of a signal from l1 are finite.
44
Initial Properties of the Discrete-Time Fourier Transform
1. The Discrete-Time Fourier Transform of a signal is continuous.
Proof.For a little increment, ΔΩ, applying the definition of the
Discrete-Time Fourier Transform, it can be written:
The absolute value of this difference is bounded.[ ] ( ) [ ]∑∑
∞
−∞=
ΔΩ−∞
−∞=
ΔΩ−Ω− ⎟⎠⎞⎜
⎝⎛ +⋅≤−⋅⋅
n
jn
n
jnjn enxeenx 11
23
45
Taking the limit for ΔΩ→0, we have:
So:
The Discrete-Time Fourier Transform of a signal is continuous.
46
2. The Discrete-Fourier Transform is periodic with period 2π.
Applying the definition:
( ) njnjnjnj eeee Ω−π−Ω−π+Ω− =⋅= 22
But:
So:
24
47
The Case of Finite Energy Signals
In this case the Discrete-Time Fourier Transform is defined using the limit in mean square:
limit in mean square l.i.m. :
48
Examples1.
25
49
2.
3.
Applying the definition:
Complex function with the absolute value and the argument:
50
( ) ( )( )2
1 ; 11 2
a sinX Arg X arctga cosa cos a
Ω⎛ ⎞Ω = Ω = ⎜ ⎟Ω −⎝ ⎠− Ω +
26
51
Discrete Time Fourier transform for periodic signals
52
The DTFT of the unitary magnitude complex exponential -expressed with the aid of distributions
DTFT for periodic signals
27
53
Fourier series decomposition of a discrete-time signal :
for a complex exponential with the frequency kΩ0:
The DTFT is linear:
54
Taking into account the definition of the Dirac’s periodic distribution:
Let on the basis of the Fourier coefficients periodicity it can be written, So the last relation becomes:
28
55
An example
The Fourier coefficients
⇒
56
The original signal and the corresponding DTFT:
29
57
Discrete-Time Fourier Transform
1. Linearity
2. Time-shifting
[ ] [ ] ( ) ( )ax n by n aX bY .+ ↔ Ω + Ω
[ ] ( )00
j nx n n e X .− Ω− ↔ Ω
58
3. Modulation in time
[ ] ( )00
j ne x n X .Ω ↔ Ω − Ω
4. Time Scaling
( ) [ ] ( )kx n X k .↔ Ω
5. Signal’s complex conjugation
[ ] ( )* *x n X .↔ −Ω
30
59
6. Time reversal
[ ] ( )x n X .− ↔ −Ω
7. Discrete-time differentiation
[ ] [ ] ( ) ( )1 1 jx n x n e X .− Ω− − ↔ − Ω
8. Signals convolution
[ ] [ ] ( ) ( )x n y n X Y .∗ ↔ Ω Ω
60
9. Adding in time
[ ] ( ) ( ) ( )201
n
jk
Xx k X .
e π− Ω=−∞
Ω↔ + π δ Ω
−∑
10. Signals product
[ ] [ ] ( ) ( ) ( ) ( )2
1 12 2
x n y n X u Y u du X Y .π
⎛ ⎞ ⎛ ⎞↔ Ω − = Ω ⊗ Ω⎜ ⎟ ⎜ ⎟π π⎝ ⎠ ⎝ ⎠∫
11. Differentiation in frequency
[ ] ( )dXnx n j .
dΩ
↔Ω
31
61
12. Real Signal Discrete-Time Fourier Transforms’ Properties
[ ] [ ] ( ) ( )[ ] ( ) [ ] ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )
;
;
; Im X
* *
p i
x n x n X X .
x n Re X x n j Im X .
X X Arg X Arg X ;
Re X Re X Im X .
= ⇒ −Ω = Ω
↔ Ω ↔ Ω
Ω = −Ω Ω = − −Ω
Ω = −Ω Ω = − −Ω
13. Parseval’s Relation[ ] [ ] ( )
( )[ ]
[ ]
[ ] [ ] ( ) ( )[ ]
[ ] [ ]
2 2
2 2
222
222
2
1 ;2
2
X 2
,
,
n -
L l
L l
x n l , x n X d
X x n .
x n , y n l , ,Y x n , y n
−π π
−π π
∞
= ∞ π
∈ = Ω Ωπ
Ω = π
∀ ∈ Ω Ω = π
∑ ∫
62
Energy Spectral Density
( ) ( )
( )
2
2
12
X
X
S X ;
W S d .π
Ω = Ω
= Ω Ωπ ∫
SX(Ω) - square of the absolute value of the Discrete-Time Fourier Transform. The energy of the considered signal is obtained integrating it.
=Energy spectral density
=Energy
32
63
Frequency Response of Discrete-Time Linear Invariant Systems
[ ] ( ) h n H .↔ Ω
The Discrete-Time Fourier Transform of the impulse response of an LTI system gives the frequency response of that system.
64
The Response of a Discrete-Time LTI system to a Periodic Discrete-Time Input
Signal
Particular example, harmonic input signal:
33
65
Fourier coefficients:
because
66
The Case of LTI Systems Described by Linear Finite Difference Equations
with Constant Coefficients
The general form of a linear finite difference equation with constant coefficients is:
Taking the Discrete-Time Fourier Transform of the both sides it results:
34
67
The frequency response of the system is obtained dividing the Discrete-Time Fourier transform of the output by the Discrete-Time Fourier Transform of the input:
68
Examplesi)
Identifying this equation with the general form:
the coefficients ak and bk are obtained. By their substitution in the general expression of the frequency response:
35
69
with:
To determine the impulse response, the frequency response must be decomposed in simple fractions:
Applying for each simple fraction the Inverse Discrete-Time Fourier Transform, the impulse response can be obtained:
70
ii) determine the impulse response of the system from the frequency response:
decomposition in simple fractions:
the Inverse Discrete-Time Fourier Transform :
36
71
iii) the impulse response and the unit step response for :
Identifying the coefficients and applying the general relation :
The Discrete-Time Fourier Transform of the unit step signal is:
The Discrete-Time Fourier Transform of the response becomes:
72
The Discrete-Time Fourier Transform of the step response :
decomposition in simple fractions :
Inverse Discrete-Time Fourier Transform ⇒ the unit step response of the considered system:
37
73
First and Second Order Discrete LTI Systems
the frequency response of a discrete LTI system is a rational function.
The two polynomials can be factorized. Each factor is a first or a second order polynomial.
74
-product in frequency domain = a convolution in time domain ⇒
first order polynomial factor ⇔ first order system
second order polynomial factor ⇔ second order system.
-knowing the behaviour in frequency of first and second order systems we can deduce the behaviour in frequency of any discrete LTI system.
This is why the first and second order systems are so important.
38
75
For M=N, the decomposition of the frequency response in simple fractions is:
76
First Order Systems[ ] [ ] [ ] ( )
[ ] [ ] [ ] [ ]1
11 , 1 . ; 1
1 . 1
j
nn
y n ay n x n a Hae
ah n a n s n n .a
− Ω
+
− − = < Ω =−
−= σ = σ
−
39
77
78
Second Order Systemsthe linear finite difference equation - constant coefficients andtwo parameters:
The frequency response:
The impulse response:
The unit step response:
40
79
( )( )
[ ] ( ) [ ]
( )( ) ( ) ( ) ( )
( )
[ ]( ) ( )
( ) [ ]
2
22 2 2 2
2 2
01 1 .
1
1 1 1 11 1 11 1 11
1 111 1
n
j
j jj
n n
.
H h n n ne
S .e ee
s n n n .
− θ
π− Ω − Ω− Ω
θ =
Ω = ⇒ = + ρ σ− ρ
ρ ρ πΩ = − − + + δ Ω
− ρ − ρ −− ρ − ρ − ρ− ρ
⎡ ⎤ρ ρ⎢ ⎥= − ρ − + ρ σ− ρ⎢ ⎥− ρ − ρ⎣ ⎦
Two Particular Cases1.
80
( )( )
[ ] ( )( ) [ ]
[ ]( ) ( )
( ) ( )( ) [ ]
2
2 2
1
1
1 ,
1 111 1
j
n
n n
.
H .e
h n n n
s n n n .
− Ω
θ = π
Ω =+ ρ
= + −ρ σ
⎡ ⎤ρ ρ⎢ ⎥= + −ρ + + −ρ σ+ ρ⎢ ⎥+ ρ + ρ⎣ ⎦
2.
41
81
Some Graphical Representations
82
( )( ) ( )
( ) ( )
22 2 2 2 2 2
2
1
4 4 1 1 4
21 2 2
H .cos cos cos cos
r sin a r cosarctg .
r cos cos r cos
Ω =ρ Ω − ρ + ρ θ Ω + − ρ + ρ θ
θ − ΩΦ Ω = −
− θ Ω + Ω
Frequency Characteristics