1. DFT properties
Note:
it is important to ensure that the DFTs are the same length
If x1(n) and x2(n) have different lengths, the shorter sequence must be padded with zeros in order to make it the same length as the longer sequence.

2. Circular shift of a Sequence
There are two way to shift the sequence from n=0 to n=N-1
Circular shift
linear shift
If a sequence is circularly shifted, the DFT is multiplied by a complex exponential

3. Time reversal of a sequence.

4. Circular convolution

5. Circular Convolution
The circular convolution is defined as

6. Example: Perform the circular convolution of the following two sequences
Solution:
x3(1) =16, x3(2)=14, x3(3) =16

9. Note: the circular convolution can be obtianed by DFT and IDFT as:
X1(k)=DFT{x1(n)}
X2(k)=DFT{x2(n)}
Then X3(k)=X1(k).X2(k) IDFT x3(n)=x1(n) x2(n)
H/w : Compare the result of DFT with the previous example N

10. LINEAR FILTERING METHODS BASED ON THE DFT
The DFT is use as a computational tool for linear system analysis and, especially, for linear filtering. possesses an output spectrum Y(K) = X(K)H(K) The frequency response H(K) The system excited with an input signal that has a spectrum X(K)

11. Linear convolution by DFT
Pad the sequences h(n) and x(n) with zeros so that they are of length N ≥ NI + N2 - 1.
Find the N -point DFTs of h(n) and x(n).
Multiply the DFTs to form the product Y (k) = H (k).X ( k ) .
Find the inverse DFT of Y(k).

12. Example : Determine the response of the FIR filter with impulse response use DFT and IDFT,
Solution: The input sequence has length L = 4 and the impulse response has length M = 3

16. Note that : There are some difficulties with the DFT approach.
Note that : There are some difficulties with the DFT approach.
if x(n) is very long , then the amount of time for computing DFTs is very long and in the process accept very long processing delays.
The solution to these problems is to use block convolution, which involves segmenting the signal to be filtered, x(n), into sections. Each section is then filtered with the FIR filter h(n), and the filtered sections are pieced together to form the sequence y(n). There are two block convolution techniques. The first is overlap-add, and the second is overlap-save.