Transcript Document

Fourier Series Summary
(From Salivahanan et al, 2002)
A periodic continuous signal f(t), - < t< , with the
fundamental frequency w0 (i.e. f(t) = f(t + 2/w0)) can be
represented as a linear combination of sinusoidal functions
with periods mw0, where mN+.
represented by sine and cosine:

1
f (t )  a0   am cosm w0t   bm sin m w0t 
2
m 1
represented by complex exponential:
f (t ) 

c
m  
m
e
jmw0t
(1)
am and bm can be resolved as
am 
bm 
 / w0
 f t cos mw t dt ,
w0

0
 / w0
 / w0
w0

m  0,1,2,...
 f t sin mw t dt ,
0
m  1,2,...
 / w0
represented by complex exponential:
w0
cm 
2
c m
w0

2
 / w0

f t e  jmw0t dt , m  0,1,2,...
 / w0
 / w0

f t e jmw0t dt , m  0,1,2,...
 / w0
cm and c-m are complex conjugate pairs
(2)
The integral can also be performed within [0, 2/w0]
w0
cm 
2
cm
w0

2
2 / w0

f t e  jmw0t dt, m  0,1,2,...
0
2 / w0

f t e jmw0t dt, m  0,1,2,...
0
More concise relation between {am, bm} and cm:
am  2 Recm ,
bm  2 Imcm 
In (1) and (2), f(t) and cm (- < t< , mZ) form a transform
pair, where (2) is referred to as a forward transform, and
(1) is an inverse transform.
 Fourier series specifies Fourier transform in situation of
periodic signals.
Frequency domain: we say that the periodic signal f(t) is
transformed to the frequency domain specified by mw0 (mZ) by
Fourier series. cm is the frequency component of f(t) with
respect to the frequency mw0.
The frequency domain (or spectrum) or a periodic continuous
signal is discrete.
 In contrast, the domain which the signal is defined is referred
to as the “time domain” or “space domain.”
The frequency components cm is a complex number. Hence, the
transform domain can be divided into two parts: magnitude and
phase.
amplitude : cm
phase : cm
Example:
f(t)
5
……
0
2
2 / w0
 f t e
w0
cm 
2


4
0
1
cm 
2
2 
2
w0
dt 
2
8
2 / w0

0
……
w0t
5
w0te  jmw0t dt
2
2
5
 jmw0t
w
te
d ( w0t )
0 2 0
w0t  2
e

5
 jm w0t  1
 j

2
2m
  jm 
 w0t 0
 jmw0t
5
 jmw0t
6
(only when
m > 0)
when m=0,
1
cm 
2
2
5
5
0 2 w0td (w0t )  2 2
w0t  2
5
1
2

 2 w0t  
2

 w0t 0
c-m is cm’s complex conjugate, so
c m
5
j
2m
 magnitude in the transform domain (in this case, the magnitude
is an even function that is symmetric to the y-axis)
|Cm|
5/2
5/2
5/4
0
w0
2w0
3w0
……
w
 phase in the transform domain: (it is an odd function)
Cm
/2
… 3w0
2w0
w0
0
w0
2w0
3w0 ……
w
/2
 The frequency domain of a Fourier transform contains both
the magnitude and phase spectra.
 High frequency (w=mw0 is large):
corresponds to the signal components that highly vary with time.
 Low frequency (w=mw0 is small)
corresponds to the signal components that slowly vary with time.
Parserval’s Theorem for Fourier series

1
f (t )  a0   am cosm w0t   bm sin m w0t 
2
m 1

1
 f (t )  a0 f (t )   am f (t ) cosm w0t   bm f (t ) sin m w0t 
2
m 1
2
w0  / w0
a0 w0  / w0
2
 f (t ) 

f (t )dt




/
w


/
w
0
0
2
4
 / w0
 / w0
w0 

am 
f (t ) cosm w0t dt  bm 
f (t ) sin m w0t dt

 / w0
 / w0
2 m 1
Remember that
am 
bm 
w0

w0

 / w0
 f t cos mw t dt ,
0
m  0,1,2,...
 / w0
 / w0
 f t sin mw t dt ,
0
 / w0
m  1,2,...
We have
w0
2
2

 a0  1  2
2


f
(
t
)


a

b
 / w0
 2  2 m m
 
m 1

 / w0
  cm
2

2
m 1
In general, Parserval’s theorem means “energy preservation.”
power of a periodic signal:
w0
2
power in the frequency domain:
   f (t )
 / w0
2
 / w0

c
m 1
2
n
Continuous Fourier Transform
Fourier series transforms a periodic continuous signal into the
frequency domain.
What will happen when the continuous signal is not periodic?
Consider the period of a signal with the fundamental frequency
being w0: (note that T specifies the fundamental period instead of the
time step defined before)
2 1
T

w0
f
A non-periodic signal can be conceptually thought of as a periodic
signal whose fundamental period T is infinite long, T .
In this case, the fundamental period w0  0.
Remember that the spectrum (in the frequency domain) of a
periodic continuous signal is discrete, specified by mw0 (mZ).
The interval between adjacent frequencies is w0.
When w0  0, we can image that the frequency becomes
continuous:
w0
cm 
2
 / w0

dw
 jmw0t
 jwt




f
t
e
dt

f
t
e
dt


2 
 / w0
(where the interval w0 becomes dw)
The summation in the inverse transform
becomes integral. Thus,
f (t ) 


 jwt
dw
 jwt
f t   
  f t e dte
2  




jmw0t
c
e
m
m  
Continuous Fourier Transform:
The forward transform:
F  jw 


f t e  jwt dt

The inverse transform:
1
f t  
2

jwt


F
jw
e
dw


The continuous Fourier transform has very similar forward and
inverse forms. The only difference is that the forward uses –j
and the inverse uses j in the complex exponential basis.
This suggests that the roles of time and frequency can be
exchanged, and some properties are symmetric to each other.
Both time and frequency domains in continuous Fourier transform
are continuous. The frequency waveform is also referred to as
the ‘spectrum’.
Many variations of forms of continuous F. T.
From Kuhn 2005
From Kuhn 2005
What is the continuous Fourier transform of a periodic
signal?
Dirac’s delta function is also called unit impulse function.
The unit impulse function (t) can be multiplied by a real number r
(or a complex number c), say r(t) (or c(t)), to represent the
delta function of different magnitudes (or magnitudes/angles).
The continuous Fourier transform of a periodic signal is a impulse
train.
F(jw)
5/2
5/2
5/4
0
w0
2w0
3w0
……
w
Relationship to the discrete Fourier series: the magnitudes of
the impulse functions are proportional to those computed from
the discrete Fourier series.
Periodic signal examples
Discrete Time Fourier Transform (DTFT)
From Fourier Series
Time domain is periodic  frequency domain is discrete
Remember that the continuous Fourier transform has similar
forms of forward and inverse transforms.
Question: When time domain is discrete, what happens in the
frequency domain?
 The frequency domain is periodic
Discrete time Fourier transform:
time domain: a discrete-time signal …, x[-2T], x[-T], x[0], x[T],
x[2T], … (T is the time step)
Energy signal and power signal
Forward transform of DTFT:
Property: Forward DTFT is periodic with the period ws = 2/T.
pf: Let r be any integers, then
Since X(ejwt) is periodic, according the Fourier series principle, it
can be expressed in the time domain as a linear combination of
complex exponentials, as the form shown in the forward DTFT.
The coefficients of the linear combination can then be computed
by finding the integral over a period:
Inverse transform
of DTFT:
If we skip the time step T (or simply T=1):
forward DTFT:
inverse DTFT:
In this case, DTFT is periodic with the period being 2.
In digital signal processing, discrete-time signals are of the main
interest, and DTFT is a main tool for analyzing such a signal.
Since the frequency spectrum repeats periodically with the
period 2, we usually consider only a finite-duration [,] (or [0,
2]).
High frequency region: The frequency nearing  or .
Low frequency region: The frequency nearing 0.
Aliasing Effect (c.f. Shenoi, 2006)
Find the relationship between the continuous Fourier transform
Xa(j) of the continuous function xa(t) and the DTFT X(ejwT).
Notation of continuous Fourier transform:
forward
inverse
Assume that the discrete-time signal x(nT) is uniformly sampled
from the analog signal xa(t) with the time step T. Apply the
inverse Fourier transform, we have
Let’s express this equation, which involves integration from =
to =, as the sum of integrals over successive intervals each
equal to one period 2/T = ws.
However, each term in this summation can be reduced to an
integral over the range /T to /T by changing a variable from
 to +2r/T:
Note that ej2rn = 1 for all integers r and n. By changing the order
of summation and integration, we have
Without loss of generality, we change the frequency variable 
to w,
Comparing the above equation with that of the inverse DTFT,
We get the desired relationship (between DTFT and continuous
Fourier transform):
This shows that DTFT of the sequence x(nT) generated by
sampling the continuous signal xa(t) with a sampling period T is
obtained by a periodic duplication of the continuous Fourier
transform Xa(jw) of xa(t) with a period 2/T = ws and scaled by T.
Because of the overlapping effect, more commonly known as
“aliasing,” there is no way of retrieving Xa(jw) from X(ejwT); in
other words, we have lost the information contained in the analog
function xa(t) when we sample it.
See figure explanation: (Note that when xa(t) is a real-number
signal, then the magnitude of its Fourier transform is an even
function, i.e., is symmetric to the y-axis).
Sampling Theorem (c.f. Shenoi, 2006)
When can we reconstruct the continuous signal from its uniform
sampling?
iff
(1) the continuous signal xa(t) is band limited – that is, if it is a
function such that its Fourier transform Xa(jw) = 0 for
|w|>2wb, where wb is a frequency bound.
(2) the sampling period T is chosen such that ws > 2 wb (where
wsT=2).
 That is, to avoid the situation of aliasing, the sampling
frequency shall be larger than twice of the highest frequency
of the continuous signal.
 Nyquist sampling theorem: fb (wb/2) is called the Nyquist
frequency, and 2fb is called the Nyquist rate.
See figures for explanation.
Continuous signal
The magnitude spectrum
of the continuous signal
Sampled signal
The spectrum of the sampled signal
Particular discrete-time signal examples
– Unit pulse (or unit sample) function (or discrete-time
impulse; impulse)
0 n  0
 n  
1 n  0
• An arbitrary sequence x[n], nZ can be represented as a
sum of scaled, delayed impulses.
xn 

 xk  n  k 
k  
• Unit step sequence
1 n  0
un  
0 n  0
Some DTFT Transform Pairs
 n  1
Timedomainshift :  n  n0   e
exponentia l sequence :
1 (  n  ) 
 jwn0
1
a un ( a  1) 
 jw
1  ae
n

 2 w  2k 
k  
DTFT Transform Pairs (continue)

1
un 
   w  2k 
 jw
1  ae
k  
n  1a un ( a  1) 
n
e
jw0 n

1
1  ae 
 jw 2

 2 w  w
k  
0
 2k 
1 0  n  M
sinwM  1 / 2  jwM / 2
xn  

e
sinw / 2
0 otherwise
DTFT Transform Pairs (continue)
1
w  wc
sin wc n
jw
 X e 
n
0 wc  w  
 
r n sin wp n  1
sin wp
cosw0 n    
1
un ( r  1) 
1  2r cos wp e  jw  r 2e  j 2 w

j
 j



e

w

w

2

k


e
 w  w0  2k 

0
k  
DTFT Theorems
• Linearity
x1[n]  X1(ejw),
x2[n]  X2(ejw)
implies that
a1x1[n] + a2x2[n]  a1X1(ejw) + a2X2(ejw)
• Time shifting
x[n]  X(ejw)
implies that
xn  nd   e
 jwnd
 
Xe
jw
DTFT Theorems (continue)
• Frequency shifting
x[n]  X(ejw)
implies that
e
jw0n

xn  X e
j ww0 

• Time reversal
x[n]  X(ejw)
If the sequence is time reversed, then x[n]  X(ejw)
DTFT Theorems (continue)
• Differentiation in frequency
x[n]  X(ejw)
implies that
dX e 
nxn   j
jw
dw
• Parseval’s theorem
x[n]  X(ejw)
implies that
E

 xn
n  
2
1

2

 X e 
jw

2
dw
Example
• Suppose we wish to find the Fourier transform of
x[n] = anu[n-5].
x1 n  a un 
n
x2 n  x1 n  5
 
1
 jw
1  ae
 
 X 1 e jw
 
Thus, X 2 e jw  e  j 5 w X 1 e jw 
 
xn  a 5 x2 n, so X e jw 
e  j 5w
1  ae jw
a 5e  j 5w
 jw
1  ae
Representation of Sequences by Discrete-time
Fourier Transforms (DTFT) (cf. Oppenheim et al. 1999)
• Fourier Representation: representing a signal by
complex exponentials.
1 
Inverse Fourier
jw jwn
xn 
X
e
e
dw
transform
2 
 
X e    xn e

jw
n  
 jwn
Fourier transform (or forward
Fourier transform)
– A signal x[n] is represented as the Fourier integral of the
complex exponentials in the range of frequencies [, ].
– The weight X(ejw) of the frequency applied in the integral can
be determined by the input signal x[n], and X(ejw) reveals
how much of each frequency is required to synthesize x[n].
Representation of sequences by
DTFT (continue)
jw
jw
jX e 




X e  X e e
jw
• The phase X(ejw) is not uniquely specified since any
integer multiple of 2 may be added to X(ejw) at any
value of w without affecting the result.
– Denote ARG[X(ejw)] to be the phase value in [, ].
– Since the frequency response of a LTI system is the Fourier
transform of the impulse response, the impulse response
can be obtained from the frequency response by applying
the inverse Fourier transform integral:
1
hn 
2
 H e e

jw
jwn
dw
Existence of discrete-time Fourier
transform pairs
• Whey they are transform pairs? Consider the integral
1
xˆ n 
2
 
 jwn

jwm

e dw
xme

 
 m  





 1

xm
 2
m  

1
Since
2

e
jwn  m 
xˆ n 

e
jwn  m 

dw

sin n  m  1 m  n
dw 

  n  m
 n  m 
0 m  n

 xm n  m  xn
m  
Conditions for the existence of
discrete-time Fourier transform pairs
• Conditions for the existence of Fourier transform pairs

of a signal:
 xn  
– Absolutely summable
n  
– Mean-square convergence:
lim


M  
 
 
X e jw  X M e jw
2
   xne
dw  0 , for X M e jw 
M
 jwn
n M
In other words, the error |X(ejw)  XM(ejw) | may not approach for
each w, but the total “energy” in the error does.
Conditions for the Existence of
DTFT transform Pairs (continue)
• Still some other cases that are neither absolutely
summable nor mean-square convergence, the Fourier
transform still exist:
• Eg., Fourier transform of a constant, x[n] = 1 for all n,
is an impulse train:
   2 w  2r 
X e jw 

r  
– The impulse of the continuous case is a “infinite heigh, zero
width, and unit area” function. If some properties are defined
for the impulse function, then the Fourier transform pair
involving impulses can be well defined too.
Symmetry Property of the Fourier
Transform
• Conjugate-symmetric sequence: xe[n] = xe*[n]
– If a real sequence is conjugate symmetric, then it is
called an even sequence satisfying xe[n] = xe[n].
• Conjugate-asymmetric sequence: xo[n] = xo*[n]
– If a real sequence is conjugate antisymmetric, then it is
called an odd sequence satisfying x0[n] =  x0[n].
• Any sequence can be represented as a sum of a
conjugate-symmetric and asymmetric sequences,
x[n] = xe[n] + xo[n], where xe[n] = (1/2)(x[n]+ x*[n])
and xo[n] = (1/2)(x[n]  x*[n]).
Symmetry Property of the Fourier
Transform (continue)
• Similarly, a Fourier transform can be decomposed
into a sum of conjugate-symmetric and antisymmetric parts:
X(ejw) = Xe(ejw) + Xo(ejw) ,
where
Xe(ejw) = (1/2)[X(ejw) + X*(ejw)]
and
Xo(ejw) = (1/2)[X(ejw)  X*(ejw)]
Symmetry Property of the Fourier
Transform (continue)
• Fourier Transform Pairs (if x[n]  X(ejw))
–
–
–
–
–
–
x*[n]  X*(e jw)
x*[n]  X*(ejw)
Re{x[n]}  Xe(ejw) (conjugate-symmetry part of X(ejw))
jIm{x[n]}  Xo(ejw) (conjugate anti-symmetry part of X(ejw))
xe[n] (conjugate-symmetry part of x[n])  XR(ejw) = Re{X(ejw)}
xo[n] (conjugate anti-symmetry part of x[n])
 jXI(ejw) = jIm{X(ejw)}
Symmetry Property of the Fourier
Transform (continue)
• Fourier Transform Pairs (if x[n]  X(ejw))
– Any real xe[n]  X(ejw) = X*(ejw) (Fourier transform is
conjugate symmetric)
– Any real xe[n]  XR(ejw) = XR(ejw) (real part is even)
– Any real xe[n]  XI(ejw) = XI(ejw) (imaginary part is odd)
– Any real xe[n]  |XR(ejw)| = |XR(ejw)| (magnitude is even)
– Any real xe[n]  XR(ejw)= XR(ejw) (phase is odd)
– xo[n] (even part of real x[n])  XR(ejw)
– xo[n] (odd part of real x[n])  jXI(ejw)
Example of Symmetry Properties
• The Fourier transform of the real sequence x[n] = anu[n]
1
for a < 1 is
jw
Xe 
1  ae jw
 
– Its magnitude is an even function, and phase is odd.
Discrete Fourier Transform (DFT)
Currently, we have investigated three cases of Fourier transform,
 Fourier series (for continuous periodic signal)
 Continuous Fourier transform (for continuous signal)
 Discrete-time Fourier transform (for discrete-time signal)
All of them have infinite integral or summation in either time or
frequency domains.
There still have another type of Fourier transform:
Consider a discrete sequence that is periodic in the time domain
(eg., it can be obtained by a periodic expansion of a finiteduration sequence, ie., we image that a finite-length sequence
repeats, over and over again, in the time domain).
Then, in the frequency domain, the spectrum shall be both
periodic and discrete, ie, the frequency sequence is also made
up of a finite-length sequence, which repeats over and over again
in the frequency domain.
Considering both the finite-length (or finite-duration) sequences in
one period of the time and frequency domains, leads to a
transform called discrete Fourier transform.
From Kuhn 2005
From Kuhn 2005
Four types of Fourier Transforms
Time domain nonperiodic
Time domain periodic
Continuous Fourier
transform (both
domains are
continuous)
Fourier series
(time domain
continuous, frequency
domain discrete)
Frequency
DTFT
domain periodic (time domain
discrete, frequency
domain continuous)
DFT/DTFS
(time domain discrete,
frequency domain
discrete, and both
finite-duration)
Frequency
domain nonperiodic