M. J. Roberts - 2/18/07
Web Appendix I - Derivations of the
Properties of the Discrete-Time Fourier
Transform
I.1 Linearity
Let z n = x n + y n where and are constants. Then
( ) (
Z F =
n= )
x n + y n e j 2 Fn = n= ( )
n= and the linearity property is
( )
( )
F
x n + y n X F + Y F .
I.2 Time Shifting and Frequency Shifting
Let z n = x n n0 . Then
( ) z n e
Z F =
j 2 Fn
x n n e
=
n= j 2 Fn
0
n= .
Let m = n n0 . Then
( ) x m e
Z F =
(
j 2 F m+ n0
)
=e
j 2 Fn0
n= x m e
j 2 Fm
=e
j 2 Fn0
( )
( )
j 2 Fn0
jn
F
F
x n n0 e
X F or x n n0 e 0 X e j .
(
)
Let Z F = X F F0 . Then
z n =
Z( F ) e
1
j 2 Fn
dF =
I-1
X( F F )e
1
0
j 2 Fn
dF .
( )
X F
n= and the time shifting property is
( )
( )
x n e j 2 Fn + y n e j 2 Fn = X F + Y F
M. J. Roberts - 2/18/07
Let = F F0 . Then
z n =
X ( ) e
(
)
j 2 + F0 n
1
d = e
j 2 F0 n
X () e
j 2 n
1
d = e
j 2 F0 n
x n and the frequency shifting property is
e
j 2 F0 n
(
)
F
x n X F F0 or e
j 2 F0 n
(
)
j F
x n X e ( 0) .
I.3 Time and Frequency Scaling
Let
x n / m , n / m an integer
z n = , otherwise
0
where m is an integer. Then
( ) z n e
Z F =
j 2 Fn
=
n= n= n/ m is an integer
x n / m e j 2 Fn
Let p = n / m , then x p = x n / m , for every integer value of p and zero for every
non-integer value of p and
( ) x p e
Z F =
j 2 Fmp
p = Therefore
( )
F
z n X mF
( )
= X mF
( )
F
or z n X e jm
I.4 Transform of a Conjugate
(
) x n e
F x n =
*
*
j 2 Fn
n= ( )
F
x * n X* F
=
x n e
n= *
+ j 2 Fn
*
= X F
( )
F
or x * n X * e j
I-2
( )
(I.1)
M. J. Roberts - 2/18/07
I.5 Differencing and Accumulation
Using the time-shifting property
(
)
( ) (
( )
) ( )
F x n x n 1 = X F e j 2 F X F = 1 e j 2 F X F
(
) ( )
(
) ( )
F
x n x n 1 1 e j 2 F X F
or
F
x n x n 1 1 e j X e j .
Let z n =
n
x m .
n
Using the fact that
m= x m = x n u n ,
m= ( )
( ) ( )
z n = x n u n Z F = X F U F .
( )
Using U F =
( )
1
1
+
F ,
1 e j 2 F 2 1
( )
X F
1
1
1
=
Z F =X F +
F
+
X
0
F
.
1
j 2 F
j 2 F
2 1
2
1 e
1 e
( )
( )
( )
() ( )
and the accumulation property of the DTFT is
( )
X F
n
x m 1 e
F
m= or
n
() ( )
1
X 0 1 F
2
( ) + X ( e ) ( ) .
X e j
x m 1 e
F
j 2 F
+
m= j0
2
j
I.6 Time Reversal
(
) x n e
F x n =
Let m = n . Then
(
) x m e
F x n =
n= + j 2 Fm
=
( )
x m e
( )
j 2 F m
( )
F
or x n X e j
I-3
( )
= X F
m= m= F
x n X F
j 2 Fn
M. J. Roberts - 2/18/07
I.7 Multiplication - Convolution Duality
Let
x m y n m .
z n = x n y n =
m= Then
( ) z n e
Z F =
j 2 Fn
=
x m y n me
j 2 Fn
.
n= m= n= Reversing the order of summation,
( ) x m y n m e
Z F =
m= j 2 Fn
(
( )
j 2 Fm
m= )
( ) x m e
Z F =Y F
x m Y ( F ) e
=
n= F y n m j 2 Fm
( ) ( )
=Y F X F .
m= (
F x m )
Therefore
( ) ( )
F
x n y n X F Y F
or
( ) ( )
F
x n y n X j Y j .
Let
z n = x n y n .
Then
( ) x n y n e
Z F =
j 2 Fn
.
n= ( ) ( X ( ) e
Z F =
n= j 2 n
1
)
d y n e
( ) X ( ) y n e
Z F =
1
)
n= (
X ( ) Y ( F ) d
1
(
j 2 F n
Y F The last integral
Therefore
j 2 Fn
=
X ( ) e
1
d =
n= j 2 n
y n e j 2 Fn d
X ( ) Y ( F ) d
1
)
is another instance of periodic convolution.
I-4
M. J. Roberts - 2/18/07
( )
( )
F
x n y n X F Y F
( )
( )
1
F
or x n y n X e j Y e j
2
(I.2)
I.8 Accumulation Definition of a Periodic Impulse
The CTFT leads to an integral definition of an impulse. In a similar manner the
DTFT leads to an accumulation definition of a periodic impulse. Begin with the definition
X(F ) =
x[n]e
and x [ n ] =
j 2 Fn
n = X(F )e
j 2 Fn
1
(I.3)
dF
Then, in
X(F ) =
x[n]e
j 2 Fn
(I.4)
n = replace x [ n ] by its integral equivalent x [ n ] = X ( ) e j 2 n d (changing F to to
1
avoid confusion between the two F’s appearing in the right-hand equation in (I.3) and in
(I.4) which have different meaning in the two equations).
X ( F ) = X ( ) e j 2 n d e j 2 Fn = 1
n = n = 0 +1
X ( ) e
j 2 ( F ) n
d .
0
or
X(F ) =
X ( ) e
j 2 ( F ) n
p
d =
X (F) e
j 2 Fn
p
n = n = or
X(F ) = Xp (F ) where
e
j 2 Fn
(I.5)
n = X ( F ) , F0 < F < F0 + 1
Xp (F) = , otherwise
0
is any arbitrary single period of X ( F ) . Since X p ( F ) is one period of X ( F ) and the
period is one, it follows that
X ( F ) = X p ( F ) 1 ( F ) .
(I.6)
Therefore, if (I.5) and (I.6) are both true that means that
e
j 2 Fn
n = I-5
= 1 ( F )
M. J. Roberts - 2/18/07
and, since 1 ( F ) is an even function,
e
= 1 ( F ) .
j 2 Fn
n = I.9 Parseval’s Theorem
The total signal energy in x n is
n= 2
x n =
X( F )e
j 2 Fn
1
n= 2
dF =
( X( F )e
j 2 Fn
1
n= dF
)( X ( ) e
1
j 2 n
)
*
d
or
2
x n =
n= X ( F ) X ( ) e
*
n= 1
(
)
j 2 F n
ddF .
1
We can exchange the order of summation and integration to yield
2
n= x n =
X ( F ) X ( ) e
*
1
1
(
(
2
n= x n =
ddF
n= = 1 F
)
j 2 F n
)
X ( F ) X ( ) ( F ) ddF
*
1
1
and
n= 2
x n =
X ( F ) X ( F ) dF = X ( F )
*
1
1
2
dF ,
proving that the total energy over all discrete-time n is equal to the total energy in one
fundamental period of DT frequency F (that fundamental period being one for any DTFT).
The equivalent result for the radian-frequency form of the DTFT is
n= 2
x n =
1
2
2
I-6
( )
X e j
2
d .