The invention will be more fully understood by reference to the following detailed description of the invention in conjunction with
In N-bit fixed-point computing, every integer value is in the range of [−2N−1, 2N−1]. If a value exceeds 2N−1−1, an overflow occurs; if a value is below −2N−1, an underflow occurs. Both overflow and underflow could be handled by requiring that the input data be sufficiently small so that the possibility of overflow/underflow is avoided. However, if the input data is small, computation precision can be sacrificed. Thus, balancing the tradeoff between overflow/underflow prevention and computation precision is an important goal in fixed-point computing. If not properly addressed, computation precision will be inferior.
As each number is represented by a finite-length sequence of binary digits, rounding (or truncation) brings in a computation error which can often be treated in terms of an additive noise. Such error is referred as a rounding error.
As shown above, the inverse discrete Fourier transform is
where
n=0,1, . . . , N−1.
It is well known to those skilled in signal processing that the discrete Fourier transform and its inverse transform can be efficiently implemented by fast Fourier transform algorithms. The presently disclosed technique is illustrated in the context of an inverse discrete Fourier transform, though the forward discrete Fourier transform is processed in an analogous fashion.
For the inverse discrete Fourier transform, when N=2l for some integer l, a decimation-in-frequency fast Fourier transform algorithm is commonly employed. The decimation-in-frequency fast Fourier transform algorithm is an iteration of a butterfly operation
where r is an exponential power, whose value depends on the locations of p and q.
In a fixed-point implementation, the dynamic range of {xm+1(p),xm+1(q)} is half of {xm(p),xm(q)}, which is undesirable for an implementation of iterative computation, as it would require different scaling factors for {xm(p),xm(q)} and for Fourier transform twiddle factors at each iteration. To keep the dynamic range of the input and output unchanged at each iteration, the following butterfly operation is applied instead
x
m+1(p)=xm(p)+xm(q),
x
m+1(q)=(xm(p)−xm(q))WNr.
The foregoing method for improving the computation precision in fast Fourier transform calculations can be implemented by a wide variety of computing hardware and software, including specially programmed general purpose computing systems, custom-designed computing hardware including application specific integrated circuits (ASICs), etc.
These and other embodiments of the invention illustrated above are intended by way of example and should not be viewed as limiting the scope of the disclosure or of the claims. The actual scope of the invention is to be limited solely by the scope and spirit of the following claims.