1. APPLICATION of the DFT: Convolution of Finite Sequences.
In filtering applications we need to implemenent a linear convolution between the input signal and the impulse response of the filter.
zero when

2. To have all sequences of the same length N-M-1, we pad them with zeros
DFT
IDFT
zero
pad
DFT

3. Convolution of Long Data Streams
Problem: in general one of the sequences (the data) is much longer than the other sequence (the impulse response of the filter). In this case we do block processing by subdividing the data into smaller sections.
There are two methods to perform this operation: Overlap and Save and Overlap and Add. L L L

4. it depends on the boundary of the block
See the convolution of every block by itself:
it depends on the boundary of the block

5. Overlap and Add.
Convolve each section and add the “tail” to the next section L L L
add
add

6. NOT affected by sectioning of the data
See a different way:
saved from
NOT affected by sectioning of the data

7. Overlap and Save
save L L L
Since we disregard the “transient” response we can just use circular convolution
discard M-1 values