FFT-based FIR filtering using overlap-add method

filters the data in vector `y`

= fftfilt(`d`

,`x`

)`x`

with a `digitalFilter`

object `d`

.

`fftfilt`

filters data using the efficient FFT-based method of
*overlap-add*
[1], a frequency domain
filtering technique that works only for FIR filters by combining successive frequency domain
filtered blocks of an input sequence. The operation performed by `fftfilt`

is described in the time domain by the difference equation:

$$y(n)=b(1)x(n)+b(2)x(n-1)+\cdots +b(nb+1)x(n-nb)$$

An equivalent representation is the Z-transform or frequency domain description:

$$Y(z)=\left(b(1)+b(2){z}^{-1}+\cdots +b(nb+1){z}^{-nb}\right)X(z)$$

`fftfilt`

uses `fft`

to implement the overlap-add method. `fftfilt`

breaks an
input sequence `x`

into length *L* data blocks, where
*L* must be greater than the filter length *N*.

and convolves each block with the filter `b`

by

y = ifft(fft(x(i:i+L-1),nfft).*fft(b,nfft));

where `nfft`

is the FFT length. `fftfilt`

overlaps
successive output sections by `n-1`

points, where `n`

is the
length of the filter, and sums them.

`fftfilt`

chooses the key parameters `L`

and
`nfft`

in different ways, depending on whether you supply an FFT length
`n`

for the filter and signal. If you do not specify a value for
`n`

(which determines FFT length), `fftfilt`

chooses
these key parameters automatically:

If

`length(x)`

is greater than`length(b)`

,`fftfilt`

chooses values that minimize the number of blocks times the number of flops per FFT.If

`length(b)`

is greater than or equal to`length(x)`

,`fftfilt`

uses a single FFT of length2^nextpow2(length(b) + length(x) - 1)

This computes

y = ifft(fft(B,nfft).*fft(X,nfft))

If you supply a value for `n`

, `fftfilt`

chooses an
FFT length, `nfft`

, of `2^nextpow2(n)`

and a data block
length of `nfft`

- `length(b)`

+ `1`

. If
`n`

is less than `length(b)`

, `fftfilt`

sets `n`

to `length(b)`

.

[1] Oppenheim, Alan V., Ronald W.
Schafer, and John R. Buck. *Discrete-Time Signal Processing*. 2nd Ed.
Upper Saddle River, NJ: Prentice Hall, 1999.

`conv`

| `designfilt`

| `digitalFilter`

| `filter`

| `filtfilt`