Discrete-Time Complex
Exponential Sequence.
x[n] = Cα n ,
where C and α are in general complex numbers.
Alternatively we can express the sequence in the
following form : βn
β
x[n] = Ce , where α = e .
Although this form is similar to the continuous - time
exponential signal we have described previously, the
former form is perferred when dealing with the
discrete - time sequence.
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Real Exponential Signal
x[n] = C *α n where α > 1 e.g .
x[n] = 2 *1.1n
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Real Exponential Signal
x[n] = C *α n where 0 < α < 1
x[n] = 2 * 0.9 n
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Real Exponential Signal
x[n] = C *α n where - 1 < α < 0
x[n] = 2 * (−0.9) n
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Real Exponential Signal
n
x[n] = C *α where α < −1
x[n] = 2 * (−1.1) n
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Real Exponential Signal
n
x[n] = C *α where α = −1
x[n] = 2 * (−1)
n
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Real Exponential Signal
Real-valued discrete exponentials
are used to describe:1) Population growth as function of
generation.
2) Total return on investment as a
function of day, month or
quarter.
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Sinusoidal Signals
x[n] = Ce βn , let C = 1 & β = jω o be purely an imaginary number.
∴ x[n] = e jω o n .
This signal is closely related to sinusoidal signal :
x[n] = A cos(ϖ o n + φ ).
Taking n as dimensionless, then both
ϖ o and φ have units of radians.
From Euler' s relation : -e
jω o n
= cos ω o n + j sin ω o n
A jφ jω o n A − jφ − jω o n
A cos(ω o n + φ ) = e e
+ e e
2
2
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Discrete-time Sinusoidal
Signals
x[n] = cos( 2πn / 12)
x[n] = cos(8πn / 31)
x[n] = cos(n / 6)
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Discrete-time Sinusoidal
Signals
These discrete-time signals possessed:1) Infinite total energy
2) Finite average power.
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General Complex Exponential
Signals
The general discrete - time complex exponential
can be interpreted in terms of real exponentials
and sinusoidal signals.
Writing C and α in polar form : jθ
C =| C | e , α =| α | e
jω o
.
x[n] = Cα n =| C || α |n cos(ω o n + θ ) + j | C || α |n sin(ω o n + θ )
| α |= 1, real & imaginary parts are sinusoidal.
| a |< 1, sinusoidal decaying exponentially,
| a |> 1, sinusoidal growing exponentially.
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General Complex Exponential
Signals
α >1
α <1
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Periodicity Properties of Discretetime Complex Exponentials
Two properties of continuous - time counterpart e
jω o t
1)The Larger is ω o , the higher is the rate of oscillation.
2) e
jω o t
is periodic for any value of ω o .
There are differences in each of the
above properties for the discrete-time
jω o n
case of e .
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Periodicity Properties of Discrete-time
Complex Exponentials
Consider the discrete - time complex exponential
with frequency ω o + 2π :
e j (ω o + 2π ) n = e j 2πn e jω o n = e jω o n .
From this we conclude that the exponential at
frequency ω o + 2π is the same as that at frequency ω o .
Similarly at frequencies ω o ± 2π , ω o ± 4π , and so on.
This is very different from the continuous - time case whereby
the signals are all distinct for all distinct values of ω o .
Because of this periodicity of 2π , we need only to consider
frequency interval of 2π in the case for discrete - time signals.
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Periodicity Properties of Discrete-time
Complex Exponentials
Because of this implied periodicity of discrete - time signal,
the signal e jωo n does not have a continually increasing rate of oscillation
as ω o is increased in magnitude.
Increasing ω o from 0 (d.c., constant sequence, no oscillation)
the oscillation increases until ω o = π , thereafter the oscillation
will decrease to 0 i.e. a constant sequence or d.c. signal at ω o = 2π .
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Periodicity Properties of Discrete-time
Complex Exponentials
Therefore, low frequencies occurs at ω o = 0, ± 2π ,
± even multiple of π .
High frequencies are at ω o = ±π ,±3π ,±odd multiple of π .
Note for ω o = π , odd multiple ofπ , e jπ n = (e jπ ) n = (−1) n ,
the signal oscillates rapidly, changing sign at each point in time.
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Periodicity Properties of Discrete-time
Complex Exponentials
x [ n ] = cos( 0 n ) = 1
x [ n ] = cos( π n / 8 )
x [ n ] = cos( π n / 4 )
x [ n ] = cos( π n )
x [ n ] = cos( π n / 2 )
x [ n ] = cos( 7 π n / 4 )
x [ n ] = cos( 3π n / 2 )
x [ n ] = cos( 15 π n / 8 )
x [ n ] = cos( 2 π n )
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Periodicity Properties of Discrete-time
Complex Exponentials
Second property concerns the
periodicity of the discrete-time
complex exponential.
In order for e jω o n to be periodic with period N > 0,
e jω o ( n + N ) = e jω o n , or equivalently e jω o N = 1.
∴ω o N must be a multiple of 2π .
ωo m
i.e. ω o N = 2π m, or equvalently
= ,
2π N
This means that the signal e jω o n is periodic if ω o / 2π
is a rational number and is not periodic otherwise.
This is also true for the discrete - time sinusoids.
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Discrete-time Sinusoidal
Signals
x[n] = cos(2πn / 12)
periodic because ω o = 2π / 12,
ωo 1
=
2π 12
x[ n] = cos(8πn / 31)
periodic because ω o = 8π / 31,
ωo 4
=
2π 31
x[n] = cos(n / 6)
not periodic because ω o = 1 / 6,
ωo
≠ rational number
2π
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Fundamental Period & Frequency of
discrete-time complex exponential
If x[n] = e
jω o n
is periodic with fundamental period N,
2π ω o
Its fundamental frequency is
=
,
N
m
The fundamental period is written as : 2π
N = m( )
ωo
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Comparison of the signal e
e jω o t
jω o t
e jω o n
Identical signals for values of ω o
Distinct signals for distinct
values ofω o .
Periodic for any choice of ω o .
Fundamental frequency ω o
and e
jω o n
separated by multiples of 2π
if ω
Periodic
only
for
integers
some
2π m
,
o
N
N > 0 and m
=
Fundamental frequency
Fundamental period
Fundamental period
ω o = 0 : undefined
2π
ωo ≠ 0 :
ωo
ω o = 0 : undefined
2π
ω o ≠ 0 : m( )
ωo
ω0
m
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Harmonically related
periodic exponential sequence
Considering periodic exponentials
with common period N samples:
φ k [ n] = e
jk ( 2π / N ) n
, for k = 0,±1,...
This set of signals possess
frequencies which are multiples of
2π / N
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Harmonically related
periodic exponential sequence
In continuous-time case
e
jk ( 2π / T ) t
are all distinct signals for k =0,±1,±2,......
φ k + N [ n] = e
=e
j ( k + N )( 2π / N ) n
jk(2π/N)n
e
j 2πn
= φ k [ n]
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Harmonically related
periodic exponential sequence
Therefore , there are only N distinct
periodic exponentials in the discrete
harmonic sequences.
φo [n] = 1, φ1[n] = e
.......φ N −1[n] = e
j 2π n / N
, φ 2 [ n] = e
j 4π n / N
,
j 2π ( N −1) n / N
Any other φ k [n] is identical to one
of the above. (e.g.φ N [n] = φ0 [n] )
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