EBU5375 Signals and systems:
Fourier series of Discrete-Time periodic signals
Dr Jesús Requena Carrión
Agenda
The notion of frequency in discrete-time signals
Fourier series representation of discrete-time periodic signals
Important properties of Fourier series
© Dr Jesús Requena Carrión
Agenda
The notion of frequency in discrete-time signals
Fourier series representation of discrete-time periodic signals
Important properties of Fourier series
© Dr Jesús Requena Carrión
Continuous-time sinusoidal signals
cos(πt)
1
1
t
cos(2πt)
1
1
t
cos(4πt)
1
1
© Dr Jesús Requena Carrión
t
Continuous-time complex exponentials
t=
e
3
4
jπt
ej2πt
Im
Im
Re
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Re
Discrete-time complex exponentials
Consider the DT complex exponential x[n] = ejΩn . This signal is:
(a) Always periodic.
(b) Never periodic.
(c) Periodic only for some values of Ω.
© Dr Jesús Requena Carrión
Discrete-time complex exponentials
Consider the discrete time complex exponentials x[n] = ejΩn . If x[n] is
periodic with period N then
x[n] = x[n + N ]
Not every value of Ω produces a periodic signal. If we assume that x[n]
is periodic,
x[n + N ] = ejΩ(n+N )
= ejΩn ejΩN
= x[n]
k
.
hence we need that ΩN = 2πk Ð→ Ω = 2π N
© Dr Jesús Requena Carrión
Discrete-time sinusoidal signals
Ω =2π(1/32)
N =32
n
Ω =2π(2/32)
N =16
n
Ω =2π(4/32)
N =8
n
© Dr Jesús Requena Carrión
Discrete-time sinusoidal signals
Ω =2π(8/32)
N =4
n
Ω =2π(16/32)
N =2
n
Ω =2π(32/32)
N =1
n
© Dr Jesús Requena Carrión
Discrete-time sinusoidal signals
Ω =2π(32 +1)/32
N =2
n
Ω =2π(32 +2)/32
N =4
n
Ω =2π(32 +4)/32
N =8
n
© Dr Jesús Requena Carrión
Discrete-time sinusoidal signals
Ω =2π(1/32)
N =32
Ω =2π(32 +1)/32
n
Ω =2π(2/32)
N =16
n
Ω =2π(32 +2)/32
n
Ω =2π(4/32)
N =8
N =4
n
Ω =2π(32 +4)/32
n
We have found some sinusoidal signals with different frequencies that are
identical. Why is that?
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N =2
N =8
n
Discrete-time complex exponentials
Ω =2π(1/32)
Ω =2π(2/32)
Ω =2π(4/32)
Ω =2π(8/32)
Ω =2π(16/32)
Ω =2π(32/32)
© Dr Jesús Requena Carrión
Discrete-time complex exponentials
ejΩn
Ω
2π 81
2π [ 18 + 1]
2π [ 18 + 2]
2π [ 18 + 3]
2π [ 18 + 4]
...
As in the case with the sinusoidal signals, we have found complex
exponentials with different frequencies that are identical.
© Dr Jesús Requena Carrión
Discrete-time complex exponentials
Consider the discrete time complex exponentials x1 [n] = ejΩ1 n and
x2 [n] = ejΩ2 n , where Ω2 = Ω1 + 2π. The angular frequency of x2 [n] is
higher than the angular frequency of x1 [n], Ω2 > Ω1 .
However, x1 [n] and x2 [n] are the same signal:
x2 [n] = ejΩ2 n
= ej(Ω1 +2π)n
= ejΩ1 n ej2πn
= ejΩ1 n
= x1 [n]
In discrete-time, all the complex exponentials that can be generated are
in any interval of frequencies of size 2π, for instance [−π, π].
© Dr Jesús Requena Carrión
Discrete-time complex exponentials
We have seen that
● In order for a complex exponential to be periodic, its angular
k
, where N is the period
frequency Ω must be such that Ω = 2π N
(whenever k and N have no factors in common).
● The frequencies Ω1 and Ω2 = Ω1 + 2π produce the same signal, since
they visit the same points in the complex plane.
Hence, we can conclude that there only exist N different complex
exponentials of period N , namely
0,
2π
1
,
N
2π
2
,
N
...
,
2π
N −1
N
For instance, Ω = 2π 2NN+2 produces the same signal as Ω = 2π N2 and
Ω = −2π N2 produces the same signal as Ω = 2π NN−2
© Dr Jesús Requena Carrión
Summary
CT complex exponentials
DT complex exponentials
Always periodic
Only periodic for Ω = 2πk/N ,
k, N integers
Different frequencies produce
different signals
Frequencies within an interval
of size 2π produce different
signals
There exist infinite complex
exponentials with period T ,
namely those of frequencies
2π
, 2 2π
,3 2π
,. . .
T
T
T
There only exist N complex
exponentials with period N ,
namely those of frequencies
2π
, 2 2π
,. . . ,N 2π
N
N
N
© Dr Jesús Requena Carrión
Agenda
The notion of frequency in discrete-time signals
Fourier series representation of discrete-time periodic signals
Important properties of Fourier series
© Dr Jesús Requena Carrión
Fourier series: basics
A Fourier series is a representation of a periodic signal as a linear
combination of harmonically related complex exponentials. By
harmonically related we mean that their frequencies can be expressed as
an integer multiple of the fundamental frequency.
For instance, in continuous-time, a periodic signal xT (t) with period T
can be expressed as
∞
∞
k=−∞
k=−∞
2π
xT (t) = ∑ ak ejkω0 t = ∑ ak ejk T
t
where ω0 = 2π/T is the fundamental frequency, kω0 are its harmonics
and ak are its coefficients.
© Dr Jesús Requena Carrión
Fourier series: basics
What’s different in discrete-time? Just the fact that there are only N
different complex exponentials with period N !
Hence, the Fourier series representation of a periodic discrete-time signals
xN [n] with period N is
2π
xN [n] = ∑ ak ejkΩ0 n = ∑ ak ejk N n
k=⟨N ⟩
k=⟨N ⟩
where Ω0 = 2π/N is the fundamental frequency, kΩ0 are its harmonics
and ak are its coefficients.
This equation is, of course, a synthesis equation.
© Dr Jesús Requena Carrión
Fourier series: Determining the coefficients I
How do we obtain the coefficients of the Fourier series of a discrete-time
periodic signal xN [n]?
One approach would be to solve the following system of N equations and
N unknowns (ak ):
xN [0] =
∑ ak
k=⟨N ⟩
xN [1] =
jkΩ
∑ ak e 0
k=⟨N ⟩
xN [2] =
...
xN [N − 1] =
jkΩ 2
∑ ak e 0
k=⟨N ⟩
jkΩ (N −1)
∑ ak e 0
k=⟨N ⟩
© Dr Jesús Requena Carrión
Fourier series: Determining the coefficients I
In matrix form, the resulting system of linear equations is:
⎡ xN [0] ⎤ ⎡1
⎢
⎥ ⎢
⎢ x [1] ⎥ ⎢1
N
⎢
⎥ ⎢
⎢
⎥ ⎢
⎢ xN [2] ⎥ = ⎢1
⎢
⎥ ⎢
⎢
⎥ ⎢⋮
⋮
⎢
⎥ ⎢
⎢xN [N − 1]⎥ ⎢1
⎣
⎦ ⎣
1
1
jΩ0
e
ejΩ0 2
j2Ω0
e
ej2Ω0 2
ejΩ0 (N −1)
ej2Ω0 (N −1)
⋯
⋱
⋯
© Dr Jesús Requena Carrión
⎤ ⎡ a0 ⎤
⎥⎢
⎥
⎥⎢ a ⎥
e
⎥⎢ 1 ⎥
⎥⎢
⎥
ej(N −1)Ω0 2 ⎥⎥ ⎢⎢ a2 ⎥⎥
⎥⎢ ⋮ ⎥
⋮
⎥⎢
⎥
⎥
j(N −1)Ω0 (N −1) ⎥ ⎢a
e
⎦ ⎣ N −1 ⎦
1
j(N −1)Ω0
Fourier series: Determining the coefficients II
Another option is using an analysis equation. Analysis equations use the
fact that harmonically related exponentials are orthogonal, so that
jk Ω n jk Ω n ∗
∑ e 1 0 (e 2 0 )
=
n=⟨N ⟩
jk Ω n −jk Ω n
∑ e 1 0 e 2 0
n=⟨N ⟩
=
j(k −k )Ω n
∑ e 1 2 0
n=⟨N ⟩
⎧
⎪
⎪N
= ⎨
⎪
0
⎪
⎩
if k1 − k2 = 0, ±N, ±2N, . . .
otherwise
© Dr Jesús Requena Carrión
Fourier series: Determining the coefficients II
So we know that
xN [n] =
jkΩ n
∑ ak e 0 ,
k=⟨N ⟩
∑ e
jk1 Ω0 n
(e
)
jk2 Ω0 n ∗
n=⟨N ⟩
⎧
⎪
⎪N
= ⎨
⎪
0
⎪
⎩
if k1 − k2 = 0, ±N, ±2N, . . .
otherwise
Let us calculate
∑ xN [n](e
)
jmΩ0 n ∗
n=⟨N ⟩
Hence
ak =
⎡
⎤
⎢
⎥
jkΩ0 n ⎥ −jmΩ0 n
⎢
=
∑ ⎢ ∑ ak e
⎥e
⎥
n=⟨N ⟩ ⎢
⎣k=⟨N ⟩
⎦
= am N
1
−jkΩ0 n
∑ xN [n]e
N n=⟨N ⟩
© Dr Jesús Requena Carrión
Fourier series of periodic signals: summary
Continuous-time, ω0 =
ak =
1
−jkω0 t
dt
∫ xT (t)e
T ⟨T ⟩
Discrete-time, Ω0 =
2π
T
1
−jkΩ0 n
∑ xN [n]e
N n=⟨N ⟩
Analysis
ak =
Synthesis
xN [n] = ∑ ak ejkΩ0 n
∞
xT (t) = ∑ ak ejkω0 t
2π
N
k=⟨N ⟩
k=−∞
© Dr Jesús Requena Carrión
Agenda
The notion of frequency in discrete-time signals
Fourier series representation of discrete-time periodic signals
Important properties of Fourier series
© Dr Jesús Requena Carrión
Parseval’s relation
The average power of a periodic signal xN [n] can be calculated in the
time domain and by using the coefficients of its Fourier series:
1
2
2
∑ ∣x[n]∣ = ∑ ∣ak ∣
N n=⟨N ⟩
k=⟨N ⟩
The coefficient ak of the Fourier series of xN [n] tells us how much of
the harmonic frequency kω0 there is in the signal.
© Dr Jesús Requena Carrión
Linearity and time shifting
Consider two periodic signals xN [n] and yn [n] with period N and
Fourier coefficients ak and bk , respectively. Then:
● The signal zN [n] = AxN [n] + ByN [n] is periodic with period N
and its Fourier coefficients are ck = Aak + Bbk .
● The signal vN [n] = xN [n − n0 ] is periodic with period N and its
Fourier coefficients are dk = ejkΩ0 n0 ak .
© Dr Jesús Requena Carrión
Fourier series and LTI systems
xN (n)
yN (n)
H(ω)
xN [n] = ∑ ak ejkΩ0 n Ð→ yN [n] = ∑ H(kΩ0 )ak ejkΩ0 n
k=⟨N ⟩
k=⟨N ⟩
Hence, the Fourier coefficients of yN [n] are bk = ak H(kΩo ). In words,
they are a filtered version of the coefficients ak .
© Dr Jesús Requena Carrión