3.1 Running fft

Some applications need DFT frequencies of the most recent $N$ samples on an ongoing basis. One example is DTMF , or touch-tone telephone dialing, in which a detection circuit must constantly monitor the line fortwo simultaneous frequencies indicating that a telephone button is depressed. In such cases, most of the data in each successive block of samples is the same,and it is possible to efficiently update the DFT value from the previous sample to compute that of the current sample. illustrates successive length-4 blocks of data for which successive DFT values may be needed.The running FFT algorithm described here can be used to compute successive DFT values at a cost of only two complex multiplies and additionsper DFT frequency.

The running FFT algorithm is derived by expressing each DFT sample, $X_{n+1}({\omega }_k)$ , for the next block at time $n+1$ in terms of the previous value, $X_n({\omega }_k)$ , at time $n$ . $$X_n({\omega }_k)=\sum_{p=0}^{N-1} x(n-p)e^{-(i{\omega }_{k}p)}$$ $$X_{n+1}({\omega }_k)=\sum_{p=0}^{N-1} x(n+1-p)e^{-(i{\omega }_{k}p)}$$ Let $q=p-1$ : $$X_{n+1}({\omega }_k)=\sum_{q=-1}^{N-2} x(n-q)e^{-(i{\omega }_k(q-1))}=e^{i{\omega }_k}\sum_{q=0}^{N-2} x(n-q)e^{-(i{\omega }_{k}q)}+x(n+1)$$ Now let's add and subtract $e^{-(i{\omega }_k(N-2))}x(n-N+1)$ :

Successive computation of a specific DFT frequency for overlapped blocks can also be thought of as a length- $N$ FIR filter . The running FFT is an efficient recursive implementation of this filter for this special case. shows a block diagram of the running FFT algorithm.The running FFT is one way to compute DFT filterbanks . If a window other than rectangular is desired, a running FFTrequires either a fast recursive implementation of the corresponding windowed, modulated impulse response, or it must have few non-zerocoefficients so that it can be applied after the running FFT update via frequency-domain convolution. DFT-symmmetric raised-cosine windows are an example.