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limitation of czt
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Limitation of CZT:


One limitation in using the CZT algorithm to evaluate the Z-transform off the unit circle stems from the fact that we may be required to compute for large number n. If  differs very much from 1.0,  can becomes large. We require large n when either M or N becomes large, since we need

To evaluate for n in the range –N<n<M. For example, if, and n=1000,  which exceeds the single precision floating point capability of most computers by a large amount. Hence the tails of the functions  can be greatly in error, thus causing the tails of the convolution (the high frequency terms) to be grossly inaccurate. The low frequency terms of the convolution will also be slightly in error but these errors are negligible in general.

The limitation on contour distance in or out from the unit circle is again due to computation of  . As  deviates significantly from 1.0, the number decreases. It is of importance to stress, however, that for   ,there is no limitation of this type since  is always of magnitude 1.

The other main limitation on the CZT algorithm stems from the fact that two L point. And one L/2 point, FFT’s must be greater than N+M-1 as mentioned previously.

We need one FFT and 2l storage locations for the transform of the product of these two transform of; and one FFT for the inverse transform of the product of these two transforms. We do not know a way of computing the transform of   either recursively or by a specific formula (except in some trivial case). Thus we must compute this transform and store it in an extra L+2 storage locations. Of course, if many transforms are to be done with the same value of L, we need not compute the transform of  each time.

We can compute the quantities  recursively as they are needed to save computation and storage. This is easily seen from fact that

   

If we define

 

 

And

   

Then

 

And

 

 

Setting A = 1 in (19) to (23) provides an algorithm for the coefficients required for the output sequence. A similar recursion formula can be obtained for generating the sequence. The user is cautioned that recursive computation of these coefficients may be a major source of numerical error, especially when  or.