Properties of Discrete-Time Fourier Series

where LHS represents the signal and RHS its DFS coefficients

1. Periodicity DFS coefficients:

As we have noted earlier that DFS Coefficients are periodic with period N.

2. Linearity of DFS:

If

If both the signals are periodic with same period N then

To prove the first equation we use equation (6.3). The DFS coefficients are given by

let n - m = l , we get

since is periodic we can use any N consecutive values, then

We can prove the relation (6.6) in a similar manner starting from equation (6.3)

4. Duality:

From equation (6.2) we see that

Thus

Interchanging the role of k and n we get

comparing this with (6.3) we see that DFS coefficients of are the original periodic sequence is reversed in time and multiplied by N. This is known as duality property. If

then

5. Complex conjugation of the periodic sequence:

substituting in equation (6.3) we get

6. Time reversal:

From equation (6.3) we have the DFS coefficient

putting m = - n we get

Since is periodic, we can use any N consecutive values

7. Symmetry properties of DFS coefficient:

In the last chapter we discussed some symmetry properties of the discrete time Fourier transform of aperiodic sequence. The same symmetry properties also hold for DFS coefficients and their derivation is also similar in style using linearity, conjugation and time reversal properties DFS coefficients.

Let us define

sequence is obtained by inserting ( m - 1) zeros between two consecutive values of. Thus Thus is also periodic, but period is mN. The DFS coefficients are given by

putting

as non zero terms occur only when r = 0

If we define then is periodic with period equal to least common multiple (LCM) of M and N. The relationship between DFS coefficients is not simple and we omit it here.

9. Difference

This follows from linearity property.

10. Accumulation

Let us define

will be bounded and periodic only if the sum of terms of over one period is zero, i.e. , which is equivalent to. Assuming this to be true

11. Periodic convolution

Let and be two periodic signals having same period N with discrete Fourier series coefficients denoted by and respectively. If we form the product then we want to find out the sequence whose DFS coefficients are. From the synthesis equation we have

substituting for in terms of we get

as inner sum can be recognized as from the synthesis equation. Thus

The sum in the equation (6.15) looks like convolution sum, except that the summation is over one period. This is known as periodic convolution. The resulting sequence is also periodic with period N. This can be seen from equation (6.15) by putting m + N instead of m.

The Duality theorem gives analogous result when we multiply two periodic sequences.

The DFS coefficients are obtained by doing periodic convolution of and and multiplying the result by 1/N. We can also prove this result directly by starting from the analysis equation. The periodic convolution has properties similar to the aperiodic (linear convolution).It is cumulative, associative and distributes over additions of two signals.

The properties of DFS representation of periodic sequence are summarized in the Table 6.1