1471 Words6 Pages

What, then, is the purpose of the DFT? A great many procedures, techniques and theorems have been developed to work with functions in the frequency domain. As it turns out, in the frequency domain we may easily perform relatively difficult mathematical techniques like differentiation, integration, or convolution via simple multiplication and division (in some cases this is the only way we can perform these operations). At a higher level of problem solving, we can perform minor miracles. We may, of course, examine the frequency spectra of time domain wave shapes, and taking the next obvious step, perform digital filtering. From here it is only a small step to enhance photographic images bringing blurry pictures into sharp focus, but we may*…show more content…*

The "Horner Scheme" takes advantage of this generalization by solving the polynomial in the following form: F(x)=A_0+〖x(A〗_1+x(A_2+x(A_3+..(xA_n )..))) (1.6) where we have repeatedly factored x out of the series at each succeeding term. Now, at the machine language level of operation, numbers are raised to an integer power by repeated multiplication, and an examination above will show that for an Nth order polynomial this scheme reduces the number of multiplications required from (N2+N)/2 to N. When one considers that N ranges upwards of 30 (for double precision functions), where the Horner Scheme yields execution times an order of magnitude faster, the power of this algorithm becomes apparent. The above is particularly prophetic in our case. The DFT, although one of the most powerful weapons in the digital signal processing arsenal, suffers from the same malady as the Taylor series described above—when applied to practical problems it tends to bog down—it takes too long to execute. The FFT, in a way that is quite analogous to the Horner Scheme just described, is an algorithm that greatly reduces the number of mathematical operations required to perform a DFT. Unfortunately the FFT is not as easy

The "Horner Scheme" takes advantage of this generalization by solving the polynomial in the following form: F(x)=A_0+〖x(A〗_1+x(A_2+x(A_3+..(xA_n )..))) (1.6) where we have repeatedly factored x out of the series at each succeeding term. Now, at the machine language level of operation, numbers are raised to an integer power by repeated multiplication, and an examination above will show that for an Nth order polynomial this scheme reduces the number of multiplications required from (N2+N)/2 to N. When one considers that N ranges upwards of 30 (for double precision functions), where the Horner Scheme yields execution times an order of magnitude faster, the power of this algorithm becomes apparent. The above is particularly prophetic in our case. The DFT, although one of the most powerful weapons in the digital signal processing arsenal, suffers from the same malady as the Taylor series described above—when applied to practical problems it tends to bog down—it takes too long to execute. The FFT, in a way that is quite analogous to the Horner Scheme just described, is an algorithm that greatly reduces the number of mathematical operations required to perform a DFT. Unfortunately the FFT is not as easy

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