Department of Electrical and Computer Engineering
From Fourier Series to Fourier Transform
tjkk
k
FekXtx
][)(
dtetxT
kXF
F
Tt
t
tjk
F
0
0
)(1
][
dtetxT
kX
F
F
F
T
T
tjk
F
2
2
)(1
][
F
tjk
F
TwhendtetxT
kX F ,)(1
][
OR
Department of Electrical and Computer Engineering
dtetxX tj )()(
Let’s Consider a function
We can express X[k] in terms of X(ω)
)(1
][ F
F
kXT
kX
F
tjk
F
TwhendtetxT
kX F ,)(1
][
Department of Electrical and Computer Engineering
t
ω
FT
X )(][kX
)(tx
2
FT
2
FT
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxT1
Department of Electrical and Computer Engineering
t
)(tx
2
FT
2
FT
ω
FT
X )(][kX
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxT2
Department of Electrical and Computer Engineering
t
)(tx
2
FT
2
FT
FT
X )(][kX
ω
F F2 F30FF2F3
Ff Ff2 Ff30FfFf2Ff3
FxTx22
Department of Electrical and Computer Engineering
t
FT
X )(
)(tx
FxT
TransformFourier
][kX
Department of Electrical and Computer Engineering
dtetxX tj )()(
Fourier Transform of a Singal x(t)
)]([)( 1 XFtx
Now Let’s talk about the Inverse Fourier Transform
)]([)( txFX
Department of Electrical and Computer Engineering
k
tjk FekXtx
][)(
k
tjk
F
F FeT
kXtx
)()(
k
tjk
F
eT
kXtx )()(
22
F
FT
k
tjkekX
tx
2
)()(
FTF
lim
Department of Electrical and Computer Engineering
k
tjkekXtx
)(2
1)(
k
tjkekXtx )(2
1)(
deXtx tj
)(2
1)(
)]([)( 1 XFtx Inverse Fourier Transform
22
F
FT
k
tjkekX
tx
2
)()(
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dtetxX tj )()(
Fourier Transform of a Signal x(t) )]([)( txFX
)()( Xtx
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dtetxX tj )()(
Fourier Transform of a Signal x(t)
)]([)( txFX
OR
dtetxfX ftj 2)()(
)]([)( txFfX
)]([)( 1 fXFtx
dfefXtx ftj
2)()(
OR
Department of Electrical and Computer Engineering
)]([)( 1 XFtx
deXtx tj
)(2
1)(
Inverse Fourier Transform
dedt
dX
dt
tdx tj
)(2
1)(
deXjdt
tdx tj
)(2
1)(
)]([)( 1 XjF
dt
tdx
)()(
Xjdt
tdx
Department of Electrical and Computer Engineering
00)( ttDefinition:
1)(
t
Fourier Transform of Impulse Function
t
)(t
dtetxX tj )()(
Fourier Transform of a Singal x(t)
1)()( 0
edtetX tj
Department of Electrical and Computer Engineering
Let’s revisit LTI system
)(tx C
R
)(ty
)(ti
)()()(
txtydt
tdyRC
)()()( XYYRCj
)(]1)[( XRCjY
1
1
)(
)(
RCjX
Y
Department of Electrical and Computer Engineering
)(tx C
R
)(ty
)(ti
1
1
)(
)(
RCjX
Y
RCjX
Y
1
1
)(
)(
RCj
H
1
1
1
)(
When the input, x(t) is unit impulse function, the output is h(t)
RCjH
1
1)(
Department of Electrical and Computer Engineering
1. An impulse function
2. A constant function (via inverse transform)
3. Complex exponential function (via inverse transform)
4. Sinosoidal Function
5. Gate Function
Some Examples of Fourier Transform
Department of Electrical and Computer Engineering
Fourier Transform of a Constant Function
t
1
dtetxX tj )()(
Fourier Transform of a Singal x(t)
dte tj
)(tx
0
j
e tj
Let’s try indirectly – let’s find the Inverse Fourier Transform of ()
deXXFtx tj
)(2
1)]([)( 1
2
1
2
1)(
2
1)]([ 01
edeF tj
)(]2
1[
F )(2]1[ F
)(2
Department of Electrical and Computer Engineering
Fourier Transform of a Complex Exponential
Let’s find the Inverse Fourier Transform of (0)
deXXFtx tj
)(2
1)]([)( 1
tjtj edeF 0
2
1)(
2
1)]([ 00
1
)(]2
[ 0
0
tj
eF
)(2][ 00
tj
eF
)( 0
0
Department of Electrical and Computer Engineering
Fourier Transform of a Sinusoidal Signal
)( 0
0
2)cos()(
00
0
tjtjee
ttx
)()(]2
[)][cos( 000
00
tjtj
eeFtF
j
eettx
tjtj
2)sin()(
00
0
)()(]2
[)][sin( 000
00
jjj
eeFtF
tjtj
)]()([ 00 j
)]()([ 00
)( 0
0
Department of Electrical and Computer Engineering
Fourier Transform of a Rectangular Function
2/10
2/12/1
2/11
)(
t
t
t
trect
t
1
2
1
2
1
dtetxXtxF tj )()()]([
2/1
2/1
)]([ dtetrectF tj
j
e
j
e
j
e jjtj
2/2/2/1
2/1
j
ee
j
e
j
e jjjj 2/2/2/2/
Department of Electrical and Computer Engineering
j
eetrectF
jj 2/2/
)]([
j
ee jj
2
][2 2/2/
)2/(
)2/sin()2/sin(
2
)2/(sin c
t
ttc
)sin()(sin
Remember?
)sin()(sin c
or
Department of Electrical and Computer Engineering
1. Linearity
2. Symmetry
3. Scaling
4. Time Shifting
5. Frequency Shifting
6. Time Differentiation
7. Convolution
Some Properties of Fourier Transform
Department of Electrical and Computer Engineering
Linearity
)()()()( 22112211 XkXktxktxk
dtetxXtxF tj )()()]([
dtetkxtkxF tj)()]([
dtetxk tj)(
)(kX
Department of Electrical and Computer Engineering
Symmetry)()( Xtx
)(2)( xtX
deXtx tj
)(2
1)(
deXtx tj
)(2
1)(
deXtx tj
)()(2
deXtx tj
)()(2
deXx j
)()(2
dtetXx jt
)()(2
dtetXx tj
)()(2
)(2)( xtX
Department of Electrical and Computer Engineering
Scaling)()( Xtx
)(1
)(a
Xa
atx
dteatxatxF tj
)()]([
dexa
atxF aj
/)(1
)]([
dadtat Let’s make a variable change
)(1
)(1 )/(
aX
adex
a
aj
Let’s assume ‘a’ is positive
Department of Electrical and Computer Engineering
Time Shifting – linear phase shift
0)()( 0
tjeXttx
dtettxttxF tj
)()]([ 00
ddttt 0Let’s make a variable change
dexttxF
tj
)(
00)()]([
deextjj
0)(
)()( 00 Xedexe
tjjtj
Department of Electrical and Computer Engineering
)()( 00
Xetxtj
Frequency Shifting - modulation
dteetxetxF tjtjtj
00 )(])([
dtetxtjtj
0)(
dtetxtj
)( 0)(
)( 0 X
Department of Electrical and Computer Engineering
Amplitude Modulation
)()( Mtm
)]()([2
1)cos()()(mod ccc MMttmts
)(tm
)cos( tc
)cos()( ttm c
Modulating signalModulated Signal
Modulating carrier
)(mod fS
)( fM
B2B2
)()( modmod Sts
cc
Department of Electrical and Computer Engineering
Amplitude Modulation – Another Method
Can we do it with square wave? If so, what else do we need?
)(tm
)cos( tc
)cos()( ttm c
Modulating signalModulated Signal
Modulating carrier
Square Wave
Department of Electrical and Computer Engineering
Amplitude Modulation – Another Method
Can we do it with square wave? If so, what else do we need?
)(tm )cos()( ttm c
Modulating signalModulated Signal
Square Wave
BPF
)(tm
)2cos( tfc
)2cos()( tftm c
Modulating signalModulated Signal
Modulating carrier
Square Wave
What is the advantage of doing so?
Department of Electrical and Computer Engineering
)( fM
2
1
B2B2
)(mod S
c2 c3 c4cc2c3c4
Bandpass Filter
Advantage: A higher multiple of carrier frequency can be chosen
)(mod fS
B2B2
B2B2c2 c3 c4cc2c3c4 c
c
Department of Electrical and Computer Engineering
Amplitude Demodulation – Synchronous
Demodulation: )(cos)()( 2 ttmtS cdem
)]2cos()()([2
1ttmtm c
)]2()2([4
1)(
2
1)( ccdem MMMS
)cos( tc
)cos()( ttm c
Modulated signalDemodulated Signal
Demodulating carrier
)(tSdem
Low Pass Filter
Department of Electrical and Computer Engineering
Amplitude Demodulation – Envelop Detection
+
-
1vCharges Discharge
CR
Time Constant = RC
c
RC
1
BRC
2
1
2v 3v
C should decay slow enoughC should charge quick enough
Department of Electrical and Computer Engineering
deXtx tj
)(2
1)(
Time Differentiation
dedt
dX
dt
tdx tj
)(2
1)(
deXjdt
tdx tj
)(2
1)(
)]([)( 1 XjF
dt
tdx
)()(
Xjdt
tdx
Department of Electrical and Computer Engineering
Convolution
dtyxtytx )()()(*)(
Then
)()( Xtx )()( Yty If and
)()()(*)( YXtytx
dtdtyxetytxF tj ])()([)](*)([
ddttyex tj ])([)(
deYx j ])([)(
Department of Electrical and Computer Engineering
deYxtytxF j ])([)()](*)([
dexY j
)()(
)()( XY
)()()(*)( YXtytx
Department of Electrical and Computer Engineering
Let’s revisit LTI system - again
)(tx C
R
)(ty
)(ti
)()()(
txtydt
tdyRC
)()()( XYYRCj
)(1
1
)(
)(
H
RCjX
Y
)()()( HXY
)(*)()( thtxtyknowwebut
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
Department of Electrical and Computer Engineering
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
)4
(sin2
1)
2(
2
1)2(
cXtx
Department of Electrical and Computer Engineering
How about FT of
)22(2 tx
jec )4
(sin
Some Properties of Fourier Transform
)2/(sin)( cX
)()( trecttx
)(sin)( fcfX
Find the Fourier Transform of the following functions
)2(),(2),2( txtxtx
22 )
2(sin)()2( jj eceXtx
)2
(sin2)(2)(2
cXtx
)4
(sin2
1)
2(
2
1)2(
cXtx
Department of Electrical and Computer Engineering
Time For Discrete Time Fourier Series
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
dtetxT
kXF
F
Tt
t
tkfj
F
0
0
)(2)(
1][
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
F
FT
f1
whereF
FN
F1
Department of Electrical and Computer Engineering
One Important Difference
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
F
FT
f1
whereF
FN
F1
nkFj Fe)(2
nFNjnkFjnFNkj FFFFF eee)(2)(2)(2
njnkFjee F 2)(2
nkFj Fe)(2
Basis functiontkfj Fe)(2
Basis function
tfTjtkfjtfTkj FFFFF eee)(2)(2)(2
tjtkfjee F 2)(2
)]2sin()2[cos()(2
njnenkFj F
)]2sin()2[cos()(2
tjtetkfj F
for all ntkfj Fe
)(2 for all t
Department of Electrical and Computer Engineering
One Important Difference … cont.
CT Exponential Fourier Series
tkfjk
k
FekXtx)(2
][)(
where
DT Exponential Fourier Series
nkFjk
k
FekXnx)(2
][][
F
FT
f1
whereF
FN
F1
nkFj Fe)(2
Basis functiontkfj Fe)(2
Basis function
Still need infinite number
of exponential functions
Need only NF exponential
Functions
nkFjNkk
kk
F
F
ekXnx)(2
10
0
][][
tkfjk
k
FekXtx)(2
][)(
Department of Electrical and Computer Engineering
DTFS vs. CTFS
Department of Electrical and Computer Engineering
Transition from DT Fourier Series to DT Fourier Transform
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
nkFjNn
NnF
F
F
F
enxN
kX)(2
2/)1(
2/
][1
][
nkFjn
nF
FenxN
kX)(2
][1
][
FN
Fnjn
n
enxFX 2][)(
Let’s define a function
When
)(1
][ F
F
kFXN
kX
Department of Electrical and Computer Engineering
Transition from DT Fourier Series to DT Fourier Transform
nkFjNnn
nnF
F
F
enxN
kX)(2
10
0
][1
][
nkFjNn
NnF
F
F
F
enxN
kX)(2
2/)1(
2/)1(
][1
][
nkFjn
nF
FenxN
kX)(2
][1
][
FN
Fnjn
n
enxFX 2][)(
Let’s define a function
When
)(1
][ F
F
kFXN
kX
DTFT
Department of Electrical and Computer Engineering
Inverse Discrete Fourier Transform
nkFjNk
k
F
F
ekXnx)(2
1
0
][][
nkFjNkk
kk
F
F
ekXnx)(2
10
0
][][
nkFjNk
k F
F F
F
eN
kFXnx
)(2)1
0
)(][
FNWhennFkj
Nk
k
eF
FkXnx
F
)(21
0 /1
)(][
Department of Electrical and Computer Engineering
FNWhennFkj
Nk
k
eF
FkXnx
F
)(21
0 /1
)(][
nFkjNk
k
eFFkXnxF
)(21
0
)(][
dFeFXnx Fnj 2
1
)(][ DT Inverse Fourier Transform
10 fromchangesFk
Department of Electrical and Computer Engineering
Fnjn
n
enxFX 2][)(
dFeFXnx Fnj 2
1
)(][
njn
n
enxX
][)(
deFXnx nj
2
)(2
1][
Discrete Time
Fourier Transform
Discrete Time
Inverse Fourier Transform
OR OR
F2Where
Department of Electrical and Computer Engineering
1. Linearity
2. Scaling
3. Time Shifting
4. Frequency Shifting
5. Time Differencing
6. Convolution
Some Properties of Discrete Fourier Transform
Department of Electrical and Computer Engineering
CTFS CTFT
DTFS DTFT
Continuous Time
Discrete Time
Continuous
Frequency
Discrete
Frequency
Relationship between Time and Frequency Domains
Department of Electrical and Computer Engineering
Discrete Frequency Continuous Frequency
CT
DT
With Examples
Department of Electrical and Computer Engineering
Parseval’s Theorem in Continuous Frequency
n
x nxE2
][
dttxEx
2)(
Energy of CT Signals Energy of DT Signals
dffX2
)(
dX2
)(2
1
1
2)( dFFX
2
2)(
2
1dX
Department of Electrical and Computer Engineering
dttxEx
2)(
dttxtx )(*)(
dtdeXtx tj ])(*2
1)[(
ddtetxX tj
])()[(*2
1
dXdXX
2
)(2
1)()(*
2
1
Department of Electrical and Computer Engineering
Parseval’s Theorem in Discrete Frequency
0
2
0
][1
Nn
x nxN
P
0
2
0
)(1
T
x dttxT
P
Power of Periodic CT Signals Power of Periodic DT Signals
k
kX2
][
0
2][
Nk
kX
Department of Electrical and Computer Engineering
tkfjk
k
FekXtx)(2
][)(
0
0
2)(2
][,, ][T
tkfj
kXTx dtekXE F
0
2)(22
][T
tkfjdtekX F
Energy of each of the above frequency component over T0
0
22][][
0
TkXdtkXT
Department of Electrical and Computer Engineering
...]]2[]1[
...]2[]1[]0[[
22
222
0, 0
XX
XXXTE Tx
...]2[]1[
...]2[]1[]0[
)2(2)(2
)2(2)(2
tfjtfj
tfjtfj
FF
FF
eXeX
eXeXX
tkfjk
k
FekXtx)(2
][)(
From conservation of energy principle
...]]2[]1[
...]2[]1[]0[[1
22
222
,
00
XX
XXXET
P Txx
Department of Electrical and Computer Engineering
...]]2[]1[
...]2[]1[]0[[1
22
222
,
00
XX
XXXET
P Txx
k
x kXP2
][
Department of Electrical and Computer Engineering
Example
Determine the power of x(t) without performing any integration
)2cos()( tfAtx F
244
222 AAAPx
tkfjk
k
FekX)(2
][
tfjtfj FF eXeX)(2)(2
]1[]1[
tfjtfj FF eA
eA )(2)(2
22
