ta rates. 3.2.4 Differences Between the FFT and the FHT The FFT is best at modelling curves and sinusoidal waveforms. The hardest curve to model with a Fourier transform is a step function, also known as a square wave, where the edges of the waveform exhibit a theoretically infinite number of sinusoids. In practice, these can be approximated but it is the 'Achilles heel' of the Fourier transform (often described as 'the Gibbs effect'). Making Telecoms Work The FHT is best at capturing square waves. The hardest curve to model with a Hadamard transform is a basic sine/cosine curve. This is intuitively consistent with matrix theory - describing square waveforms by putting numbers into squares. Hadamard transforms when implemented as fast Walsh Hadamard transforms use only additions and subtractions and are therefore computationally efficient. Fourier transforms require many multiplications and are slow and expensive to execute. Fast Fourier transforms employ imperfect 'twiddle factors' SO trade accuracy against complexity and 'convergence delay'. The magnitude of an FFT is invariant to phase shifts in the signal. This is not true in the case of the FHT because a circular shift in one row of the Hadamard matrix does not leave it orthogonal in other rows. This is the Achilles heel of the FHT and is a weakness that underpins the ultimate limitations of CDMA in terms of error performance and susceptibility to AM/PM distortion. However, with this proviso, the Hadamard transform has been and remains a fundamental part of the signal processing chain in present mobile phones both in terms of its application in discrete processes such as channel coding and code division multiplexing but also in a support role to other processes. The fact that it is simpler to execute and has different but complementary properties makes it a useful companion to the FFT and the two processes working together provide the basis for future performance gain. Having mastered the theory, let's examine how the Hadamard transform is applied in present