Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 603

Solution

The DTFT ofδM[n] is given by

(^1) M(ejω)=

∑∞

m=−∞

F(δ[n−mM])=

∑∞

m=−∞

e−jωmM (10.32)

An equivalent result can be obtained if before we find the DTFT we find the Fourier series ofδM[n].

The coefficients of the Fourier series ofδM[n] are

1

M

M∑− 1

n= 0

δM[n]e−j^2 πnk/M=

1

M

M∑− 1

n= 0

δ[n]e−j^2 πnk/M=

1

M

so that the Fourier series ofδM[n] is

δM[n]=

M∑− 1

k= 0

1

M

ej^2 πnk/M

and its DTFT is then given by

(^1) M(ejω)=

M∑− 1

k= 0

F

(

1

M

ej^2 πnk/M

)

=

2 π

M

M∑− 1

k= 0

δ

(

ω−

2 πk

M

)

−π≤ω < π (10.33)

Notice the equivalence of Equations (10.32) and (10.33).

Putting the pairδM[n] and (^1) M(ejω)together gives an interesting relation:
∑∞
m=−∞
δ[n−mM]⇔
2 π
M

M∑− 1

k= 0

δ

(

ω−

2 πk

M

)

π≤ω < π

Both terms are discrete in time and in frequency, and both are periodic. The period ofδM[n] is

Mand the one of1(ejω)is 2π/M. Furthermore, the DTFT of an impulse train in time is also an

impulse train in frequency. However, it should be observed that the delta functionsδ[n−mM] on

the left term are discrete, while the ones on the right term,δ(ω− 2 πk/M), are continuous. n

Computation of the Fourier Series using MATLAB
Given that periodic signals only have discrete frequencies, and that the Fourier series coefficients
are obtained using summations, the computation of the Fourier series can be implemented with a
frequency discretized version of the DTFT, or the DFT, which can be efficiently computed using the
FFT algorithm. To illustrate this using MATLAB, consider three different signals as indicated in the
following script and displayed in Figure 10.9.

Each of the signals is periodic and we consider 10 periods to compute their FFTs, which provides
the Fourier series coefficients (only 3 periods are displayed in Figure 10.9). A very important issue to
remember is that one needs to input exactly one or more periods, and that the FFT length must be