Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 609

For the circular representation ofx[n−1] we shift the termx[0] from the E to the S in the clockwise

direction and shift the others the same angle to get the circular representation ofx[n−1]—

a circular shift of one. Shifting linearly by one, we obtain the equivalent representation of

x[n−1]. n

Periodic Convolution
Consider then the multiplication of two periodic signalsx[n] andy[n] of the same periodN. The
productv[n]=x[n]y[n] is also periodic of periodN, and its Fourier series coefficients are

V[m]=

N∑− 1

k= 0

X[k]Y[m−k] 0 ≤m≤N− 1

as we will show next. Thatv[n] is periodic of periodNis clear. Its Fourier series is found by letting
the fundamental frequency beω 0 =^2 Nπand replacing the given Fourier coefficients,

v[n]=

N∑− 1

m= 0

V[m]ejω^0 nm=

N∑− 1

m= 0

N∑− 1

k= 0

X[k]Y[m−k]ejω^0 nm

=

N∑− 1

k= 0

X[k]

(N− 1

∑

m= 0

Y[m−k]ejω^0 n(m−k)

)

ejω^0 kn=

N∑− 1

k= 0

X[k]y[n]ejω^0 kn=y[n]x[n]

Thus, we have that the Fourier series coefficients of the product of two periodic signals of the same
period give

x[n]y[n]⇔

N∑− 1

k= 0

X[k]Y[m−k] (10.36)

and by duality

N∑− 1

k= 0

x[k]y[n−k]⇔NX[k]Y[k] (10.37)

Although

N∑− 1

k= 0

x[k]y[n−k] and

N∑− 1

k= 0

X[k]Y[m−k]

look like the convolution sums we had before, the periodicity of the sequences makes them different.
These are calledperiodic convolution sums. Given the infinite support of periodic signals, the convolu-
tion sum of periodic signals does not exist—it would not be finite. The periodic convolution is done
only for a period of the signals.