Properties of the discrete Fourier transform

Since discrete Fourier transform is similar to the discrete Fourier series representation, the properties are similar to DFS representation. We use the notation

1. Linearity

If two finite length sequence have length M and N , we can consider both of them with length greater than or equal to maximum of M and N. Thus if

then

where all the DFTs are N-point DFT. This property follows directly from the equation (6.20)

2. Circular shift of a sequence

If we shift a finite length sequence of length N , we face some difficulties. When we shift it in right direction the length of the sequence will becam according to definition. Similarly if we shift it left , if may no longer be a finite length sequence as may not be zero for n < 0. Since DFT coefficients are same as DFS coefficients, we define a shift operation which looks like a shift of periodic sequence. From we get the periodic sequence defined by

We can shift this sequence by m to get

Now we retain the first N values of this sequence

This operation is shown in figure below for m = 2, N = 5.

Fig 6.1

We can see that is not a shift of sequence. Using the propertiesof the modulo arithmetic we have

and

The shift defined in equation (6.23) is known as circular shift. This is similar to a shift of sequence in a circular register.

Fig 6.2

3. Shift property of DFT

From the definition of the circular shift, it is clear that it corresponds to linear shift of the associated periodic sequence and so the shift property of the DFS coefficient will hold for the circular shift. Hence

and

4. Duality

We have the duality for the DFS coefficient given by , retaining one period of the sequences the duality property for the DFT coefficient will become

5. Symmetry properties

We can infer all the symmetry properties of the DFT from the symmetry properties of the associated periodic sequence and retaining the first period. Thus we have

and

We define conjugate symmetric and anti-symmetric points in the first period 0 to N - 1 by

Since

the above equation similar to

and are referred to as periodic conjugate symmetric and periodic conjugate anti-symmetric parts of. In terms if these sequence the symmetric properties are

6. Circular convolution

We saw that multiplication of DFS coefficients corresponds of periodic convolution of the sequence. Since DFT coefficients are DFS coefficients in the interval, , they will correspond to DFT of the sequence retained by periodically convolving associated periodic sequences and retaining their first period.

Periodic convolution is given by

using properties of the modulo arithmetic

and then

Since we get

The convolution defined by equation (6.28) is known as N-point-circular convolution of sequence and , where both the sequence are considered sequence of length N. From the periodic convolution property of DFS it is clear that DFT of is. If we use the notation to denote the N point circular convolution we see that

In view of the duality property of the DFT we have

Properties of the Discrete Fourier transform are summarized in the table 6.2