Continuous-Time Fourier
Transform
Continuous-Time Fourier
Transform
• Definition – The CTFT of a continuoustime signal xa (t ) is given by
• Definition – The inverse CTFT of a Fourier
transform X a ( jΩ) is given by
1 ∞
jΩt
x a (t ) =
∫ X a ( jΩ)e dΩ
2 π −∞
∞
X a ( jΩ) = ∫ xa (t )e − jΩt dt
−∞
• Often referred to as the Fourier spectrum or
simply the spectrum of the continuous-time
signal
• Often referred to as the Fourier integral
• A CTFT pair will be denoted as
CTFT
x a ( t ) ↔ X a ( jΩ )
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2
Continuous-Time Fourier
Transform
Continuous-Time Fourier
Transform
• The quantity X a ( jΩ) is called the
magnitude spectrum and the quantity θa (Ω)
is called the phase spectrum
• Both spectrums are real functions of Ω
• In general, the CTFT X a ( jΩ) exists if xa (t )
satisfies the Dirichlet conditions given on
the next slide
• Ω is real and denotes the continuous-time
angular frequency variable in radians
• In general, the CTFT is a complex function
of Ω in the range − ∞ < Ω < ∞
• It can be expressed in the polar form as
X a ( j Ω ) = X a ( jΩ ) e jθ a ( Ω )
where
θa (Ω) = arg{X a ( jΩ)}
3
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4
Continuous-Time Fourier
Transform
Copyright © 2005, S. K. Mitra
Continuous-Time Fourier
Transform
• If the Dirichlet conditions are satisfied, then
Dirichlet Conditions
• (a) The signal xa (t ) has a finite number of
discontinuities and a finite number of
maxima and minima in any finite interval
• (b) The signal is absolutely integrable, i.e.,
∞
jΩt
∫ X a ( jΩ)e dΩ
−∞
converges to xa (t ) at values of t except at
values of t where xa (t ) has discontinuities
1
2π
∞
• It can be shown that if xa (t ) is absolutely
integrable, then X a ( jΩ) < ∞ proving the
existence of the CTFT
∫ xa (t ) dt < ∞
−∞
5
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6
Copyright © 2005, S. K. Mitra
1
Energy Density Spectrum
Energy Density Spectrum
• The total energy E x of a finite energy
continuous-time complex signal xa (t ) is
given by
∞
• Interchanging the order of the integration
we get
∞
2
E x = ∫ xa (t ) dt = ∫ xa (t ) x*a (t ) dt
−∞
7
−∞
• The above expression can be rewritten as
∞
⎡ ∞
⎤
E x = ∫ xa (t )⎢ 1 ∫ X a* ( jΩ)e − jΩt dΩ ⎥ dt
2π
⎣ −∞
⎦
−∞
Copyright © 2005, S. K. Mitra
∞
2
∞
2
1
∫ x(t ) dt = 2 π ∫ X a ( jΩ) dΩ
−∞
−∞
• The above relation is more commonly
known as the Parseval’s relation for finiteenergy continuous-time signals
1
2π
=
1
2π
=
8
1
2π
∞
⎡∞
− jΩt ⎤
dt ⎥ dΩ
∫ X a* ( jΩ) ⎢ ∫ xa (t )e
⎦
⎣−∞
−∞
∞
∫ X a* ( jΩ) X a ( jΩ)dΩ
−∞
∞
2
∫ X a ( jΩ) dΩ
−∞
Copyright © 2005, S. K. Mitra
Energy Density Spectrum
• Hence
Ex =
Energy Density Spectrum
• The quantity X a ( jΩ) 2 is called the energy
density spectrum of xa (t ) and usually
denoted as
2
S xx (Ω) = X a ( jΩ)
• The energy over a specified range of
frequencies Ω a ≤ Ω ≤ Ωb can be computed
using
Ωb
E x,r = 1 ∫ S xx (Ω) dΩ
2π
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10
Ωa
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Band-limited Continuous-Time
Signals
Band-limited Continuous-Time
Signals
• A full-band, finite-energy, continuous-time
signal has a spectrum occupying the whole
frequency range − ∞ < Ω < ∞
• A band-limited continuous-time signal has a
spectrum that is limited to a portion of the
frequency range − ∞ < Ω < ∞
• An ideal band-limited signal has a spectrum
that is zero outside a finite frequency range
Ω a ≤ Ω ≤ Ωb , that is
⎧0, 0 ≤ Ω < Ω a
X a ( jΩ ) = ⎨
⎩0, Ωb < Ω < ∞
11
Copyright © 2005, S. K. Mitra
• However, an ideal band-limited signal
cannot be generated in practice
12
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2
Band-limited Continuous-Time
Signals
Band-limited Continuous-Time
Signals
• Band-limited signals are classified
according to the frequency range where
most of the signal’s is concentrated
• A lowpass, continuous-time signal has a
spectrum occupying the frequency range
Ω ≤ Ω p < ∞ where Ω p is called the
bandwidth of the signal
• A highpass, continuous-time signal has a
spectrum occupying the frequency range
0 < Ω p ≤ Ω < ∞ where the bandwidth of
the signal is from Ω p to ∞
• A bandpass, continuous-time signal has a
spectrum occupying the frequency range
0 < Ω L ≤ Ω ≤ Ω H < ∞ where Ω H − Ω L is
the bandwidth
13
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Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• Definition - The discrete-time Fourier
transform (DTFT) X (e jω ) of a sequence
x[n] is given by
• Definition - The discrete-time Fourier
transform (DTFT) X (e jω ) of a sequence
x[n] is given by
∞
X ( e jω ) =
X (e jω ) = ∑ x[n]e − jω n
n = −∞
n = −∞
j
ω
X (e ) is
X ( e jω )
15
• In general,
is a complex function
of the real variable ω and can be written as
X (e jω ) = X re (e jω ) + j X im (e jω )
Copyright © 2005, S. K. Mitra
16
• In general,
a complex function
of the real variable ω and can be written as
X (e jω ) = X re (e jω ) + j X im (e jω )
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• X re (e jω ) and X im (e jω ) are, respectively,
the real and imaginary parts of X (e jω ), and
are real functions of ω
• X (e jω ) can alternately be expressed as
X (e jω ) = X (e jω ) e jθ(ω)
where
θ(ω) = arg{ X (e jω )}
17
Copyright © 2005, S. K. Mitra
∞
∑ x[n]e− jω n
•
•
•
•
X (e jω ) is called the magnitude function
θ(ω) is called the phase function
Both quantities are again real functions of ω
In many applications, the DTFT is called
the Fourier spectrum
• Likewise, X (e jω ) and θ(ω) are called the
magnitude and phase spectra
18
Copyright © 2005, S. K. Mitra
3
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• For a real sequence x[n], X (e jω ) and X re (e jω )
are even functions of ω, whereas,θ(ω)
and X im (e jω ) are odd functions of ω
• Note: X (e jω ) = X (e jω ) e jθ(ω+ 2πk )
• Unless otherwise stated, we shall assume
that the phase function θ(ω) is restricted to
the following range of values:
− π ≤ θ(ω) < π
called the principal value
= X (e jω ) e jθ(ω)
for any integer k
•
The phase function θ(ω) cannot be
uniquely specified for any DTFT
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20
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• The DTFTs of some sequences exhibit
discontinuities of 2π in their phase
responses
• An alternate type of phase function that is a
continuous function of ω is often used
• It is derived from the original phase
function by removing the discontinuities of
2π
• The process of removing the discontinuities
is called “unwrapping”
• The continuous phase function generated by
unwrapping is denoted as θc (ω)
• In some cases, discontinuities of π may be
present after unwrapping
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22
Discrete-Time Fourier
Transform
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
• Its DTFT is given by
• Example - The DTFT of the unit sample
sequence δ[n] is given by
∞
∞
∞
X (e jω ) = ∑ α nµ[n]e− jωn = ∑ α ne− jωn
n = −∞
− jω n
∆(e jω ) = ∑ δ[n]e − jωn = δ[0] = 1
n = −∞
∞
= ∑ (α e
• Example - Consider the causal sequence
x[n] = α nµ[ n],
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Copyright © 2005, S. K. Mitra
n =0
)
as α e− jω = α < 1
α <1
Copyright © 2005, S. K. Mitra
n =0
= 1 − jω
1− α e
24
Copyright © 2005, S. K. Mitra
4
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• The DTFT X (e jω ) of a sequence x[n] is a
continuous function of ω
• It is also a periodic function of ω with a
period 2π:
• The magnitude and phase of the DTFT
X (e jω ) = 1 /(1 − 0.5 e − jω ) are shown below
0.6
2
Phase in radians
Magnitude
0.4
1.5
1
0.2
∞
X (e j ( ωo + 2 πk ) ) = ∑ x[ n]e − j ( ωo + 2 πk ) n
0
-0.2
-0.4
0.5
-3
-2
-1
0
ω/π
1
2
3
-3
-2
-1
X ( e jω ) = X ( e − jω )
25
n = −∞
∞
0
ω/π
1
2
3
θ( ω) = − θ( − ω)
Copyright © 2005, S. K. Mitra
= ∑ x[n]e
n = −∞
n = −∞
Copyright © 2005, S. K. Mitra
• Proof:
x[ n] =
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
• The order of integration and summation can
be interchanged if the summation inside the
brackets converges uniformly, i.e. X (e jω )
exists
1 π⎛ ∞
− j ωl ⎞ j ω n
• Then
⎟e dω
∫ ⎜ ∑ x[l]e
2π − π ⎝ l = −∞
⎠
29
• Now
sin π( n − l) 1, n = l
=
π( n − l )
0, n ≠ l
= δ[n − l]
• Hence
∞
∑ x[l]
∞
1 π jω( n−l ) ⎞
sin π(n − l)
x[l]⎛⎜
dω ⎟ = ∑ x[l]
∫e
2
π( n − l )
π
l = −∞
l
=
−∞
⎠
⎝ −π
l = −∞
∞
∑
Copyright © 2005, S. K. Mitra
1 π⎛ ∞
− jωl ⎞ jω n
⎟e dω
∫ ⎜ ∑ x[l]e
2π − π ⎝ l = −∞
⎠
28
Discrete-Time Fourier
Transform
=
n = −∞
• Inverse discrete-time Fourier transform:
1 π
jω jω n
x[n] =
∫ X (e )e dω
2π − π
∞
Copyright © 2005, S. K. Mitra
∞
= ∑ x[n]e − jωon = X (e jωo )
Discrete-Time Fourier
Transform
X (e jω ) = ∑ x[n]e − jω n
represents the Fourier series representation
of the periodic function
• As a result, the Fourier coefficients x[n] can
be computed from X (e jω ) using the Fourier
integral
1 π
jω jωn
x[ n] =
∫ X (e )e dω
2
π
−π
27
e
26
Discrete-Time Fourier
Transform
• Therefore
− jωo n − j 2 π k n
30
∞
sin π(n − l)
= ∑ x[l]δ[ n − l] = x[ n]
π(n − l)
l = −∞
Copyright © 2005, S. K. Mitra
5
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• Then for uniform convergence of X (e jω ) ,
• Convergence Condition - An infinite
series of the form
∞
jω
X (e ) = ∑ x[n]e
lim X (e jω ) − X K (e jω ) = 0
K →∞
− jω n
n = −∞
• Now, if x[n] is an absolutely summable
sequence, i.e., if
may or may not converge
• Let
K
X K (e jω ) = ∑ x[n]e− jω n
∞
∑ x[n] < ∞
n = −∞
n=− K
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32
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• Example - The sequence x[n] = α nµ[ n] for
α < 1 is absolutely summable as
∞
∞
1
n
n
<∞
∑ α µ[n] = ∑ α =
1− α
n = −∞
n =0
• Then
∞
∞
n = −∞
n = −∞
X (e jω ) = ∑ x[ n]e − jωn ≤ ∑ x[n] < ∞
33
for all values of ω
• Thus, the absolute summability of x[n] is a
sufficient condition for the existence of the
DTFT X (e jω )
Copyright © 2005, S. K. Mitra
and its DTFT X (e jω ) therefore converges
to 1 /(1 − α e− jω ) uniformly
34
Discrete-Time Fourier
Transform
• Since
∞
• Example - The sequence
2
1 / n, n ≥ 1
x[n] = ⎧⎨
⎩ 0, n ≤ 0
has a finite energy equal to
∞ 1 2 π2
E x = ∑ ⎛⎜ ⎟⎞ =
6
n =1 ⎝ n ⎠
• But, x[n] is not absolutely summable
an absolutely summable sequence has
always a finite energy
• However, a finite-energy sequence is not
necessarily absolutely summable
35
Copyright © 2005, S. K. Mitra
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
⎛ ∞
⎞
∑ x[n] ≤ ⎜ ∑ x[n] ⎟ ,
⎝ n = −∞ ⎠
n = −∞
2
Copyright © 2005, S. K. Mitra
36
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6
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• To represent a finite energy sequence x[n]
that is not absolutely summable by a DTFT
X (e jω ) , it is necessary to consider a meansquare convergence of X (e jω ):
π
• Here, the total energy of the error
X ( e jω ) − X K ( e jω )
must approach zero at each value of ω as K
goes to ∞
• In such a case, the absolute value of the
error X (e jω ) − X K (e jω ) may not go to
zero as K goes to ∞ and the DTFT is no
longer bounded
2
lim ∫ X (e jω ) − X K (e jω ) dω = 0
K →∞ − π
where
K
X K (e jω ) = ∑ x[n] e − jω n
n=− K
37
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38
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
• The inverse DTFT of H LP (e jω ) is given by
1 ωc jω n
hLP [n] =
∫ e dω
2π −ωc
• Example - Consider the DTFT
⎧1, 0 ≤ ω ≤ ωc
H LP (e jω ) = ⎨
⎩0, ωc < ω ≤ π
shown below
H LP (e
jω
− π − ωc
0
)
π
ωc
1 ⎛⎜ e jωc n e − jωc n ⎟⎞ sin ωc n
−
=
, −∞ < n< ∞
πn
2π ⎜⎝ jn
jn ⎟⎠
• The energy of hLP [n] is given by ωc / π
hLP [n] is a finite-energy sequence,
•
but it is not absolutely summable
40
=
1
39
Discrete-Time Fourier
Transform
ω
Copyright © 2005, S. K. Mitra
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• As a result
K
∑ hLP [n] e
n=− K
− jω n
• The mean-square convergence property of
the sequence hLP [n] can be further
illustrated by examining the plot of the
function
K sin ω n
c e − jω n
H LP , K (e jω ) = ∑
πn
n= − K
sin ωc n − jω n
e
πn
n= − K
K
= ∑
does not uniformly converge to H LP (e jω )
for all values of ω, but converges to H LP (e jω )
in the mean-square sense
41
Copyright © 2005, S. K. Mitra
for various values of K as shown next
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Copyright © 2005, S. K. Mitra
7
Discrete-Time Fourier
Transform
1
1
0.8
0.8
0.6
0.4
0.2
0.4
0
0.2
0.4
0.6
0.8
1
0
0.2
1
1
0.8
0.8
0.6
0.4
0.6
0.8
1
0.6
0.4
0.2
0.2
0
0
0
0.4
ω/π
N = 40
Amplitude
Amplitude
ω/π
N = 30
43
• As can be seen from these plots, independent
of the value of K there are ripples in the plot
of H LP , K (e jω ) around both sides of the
point ω = ωc
• The number of ripples increases as K
increases with the height of the largest ripple
remaining the same for all values of K
0.6
0.2
0
0
Discrete-Time Fourier
Transform
N = 20
Amplitude
Amplitude
N = 10
0.2
0.4
0.6
0.8
0
1
0.2
0.4
Copyright
©0.62005, S.0.8K. Mitra1
44
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• As K goes to infinity, the condition
π
2
lim ∫ H LP (e jω ) − H LP , K (e jω ) dω = 0
K →∞ − π
holds indicating the convergence of H LP , K (e jω )
to H LP (e jω )
• The oscillatory behavior of H LP , K (e jω )
approximating H LP (e jω ) in the meansquare sense at a point of discontinuity is
known as the Gibbs phenomenon
45
Copyright © 2005, S. K. Mitra
46
Discrete-Time Fourier
Transform
∞
∞
∆ →0 − ∞
−∞
lim ∫ p∆ (ω)dω = ∫ δ(ω)dω
1
∆
• The DTFT can also be defined for a certain
class of sequences which are neither
absolutely summable nor square summable
• Examples of such sequences are the unit
step sequence µ[n], the sinusoidal sequence
cos(ωo n + φ) and the exponential sequence Aα n
• For this type of sequences, a DTFT
representation is possible using the Dirac
delta function δ(ω)
Copyright © 2005, S. K. Mitra
Discrete-Time Fourier
Transform
• A Dirac delta function δ(ω) is a function of
ω with infinite height, zero width, and unit
area
• It is the limiting form of a unit area pulse
function p∆ (ω) as ∆ goes to zero satisfying
47
Copyright © 2005, S. K. Mitra
ω/π
ω/π
• Example - Consider the complex exponential
sequence
x[n] = e jωon
• Its DTFT is given by
p∆ (ω )
∞
X (e jω ) = ∑ 2πδ(ω − ωo + 2π k)
k = −∞
−∆ 0 ∆
2
2
ω
Copyright © 2005, S. K. Mitra
48
where δ(ω) is an impulse function of ω and
− π ≤ ωo ≤ π
Copyright © 2005, S. K. Mitra
8
Discrete-Time Fourier
Transform
Discrete-Time Fourier
Transform
• The function
• Thus
∞
X (e jω ) = ∑ 2πδ(ω − ωo + 2π k)
x[ n] =
k = −∞
is a periodic function of ω with a period 2π
and is called a periodic impulse train
jω
• To verify that X (e ) given above is
jω n
indeed the DTFT of x[n] = e o we
compute the inverse DTFT of X (e jω )
49
Copyright © 2005, S. K. Mitra
π
= ∫ δ(ω − ωo )e jωn dω = e jωon
−π
where we have used the sampling property
of the impulse function δ(ω)
50
Commonly Used DTFT Pairs
Sequence
1 ↔
DTFT Properties
↔
• There are a number of important properties
of the DTFT that are useful in signal
processing applications
• These are listed here without proof
• Their proofs are quite straightforward
• We illustrate the applications of some of the
DTFT properties
∞
∑ 2πδ(ω + 2π k)
k = −∞
∞
∑ 2πδ(ω − ωo + 2π k)
k = −∞
∞
1
+ ∑ π δ ( ω + 2 π k)
− jω
1− e
k = −∞
1
µ[n], ( α < 1) ↔
1 − α e − jω Copyright © 2005, S. K. Mitra
µ[n] ↔
51
52
Table 3.1: DTFT Properties:
Symmetry Relations
53
Copyright © 2005, S. K. Mitra
DTFT
δ[n] ↔ 1
e jωon
1 π ∞
jωn
∫ ∑ 2πδ(ω − ωo + 2π k)e dω
2π − π k = −∞
x[n]: A complex sequence
Copyright © 2005, S. K. Mitra
Copyright © 2005, S. K. Mitra
Table 3.2: DTFT Properties:
Symmetry Relations
54
x[n]: A real sequence
Copyright © 2005, S. K. Mitra
9
Table 3.4:General
3.4:General Properties of
DTFT
55
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10