How to efficiently update a discrete time FFT?

I have an application that needs to be optimized for speed. It's a realtime app that is required to perform a number of FFTs at relatively fast discrete time interval. The transform inputs are discretely sampled time histories of real time physical processes. Therefore the time history for a given interval is the same as the previous accept it is shifted one position and one end point is lost and one is point is added. I know there must be a straight forward way to implement a faster FFT. Can somebody point me in the right direction?

Answer 1

[Matt J]

If you are doing fft(x), I assume that x(1)=x_old gets deleted and x(N) gets replaced by x_new

%pre-computations, do these only once
    c=exp(pi*2i*(0:N-1)/N);
    X=fft(x0); %inital FFT
%end of pre-computations

X = (X +(x_new - x_old)).*c;  %update the FFT

[James]

I love it when things are easy. Thanks!