The poorman´s transform: approximating the Fourier transform without multiplication

A time-domain to frequency-domain transformation for sampled signals which is computed with only additions and trivial complex multiplications is described. This poorman´s transform is an approximation to the usual Fourier transform, obtained by quantizing the Fourier coefficients to the four values {±1, ±j}, and is especially useful when multiplication is expensive. For the general case of an N-point quantization, an analytic formula is given for the error in the approximation, which involves only contributions from aliased harmonics. Continuous-time signals are considered; in this case the approximation is exact for bandlimited signals