Fast 2-D discrete cosine transform - Vetterli - Very fast Fourier transform algorithms hard-ware for implementation. On the multiplicative complexity of the discrete Fourier transform - Winograd - Algorithms meeting the lower bounds on the multiplicative complexity of length-2 n DFTs and their connection with practical algorithm - Duhamel - Fast one-dimensional digital convolution by multidimensional techniques - Agarwal, Burrus - Improved fourier and hartley transform algorithms: Application to cyclic convolution of real data - Duhamel, Vetterli - A radix-8 fast Fourier transform subroutine for real-valued series - Bergland - An improved digit-reversal permutation algorithm for the fast Fourier and Hartley transforms - Evans - A Fast Fourier Transform algorithm using base 8 iterations - Bergland - Direct fast Fourier transform of bivariant functions - Rivard - Fast computation of discrete Fourier transforms using polynomial transforms - Nussbaumer, Quandalle - Split vector-radix 2D fast Fourier transform - Pei, Wu - Efficient Fourier transform and convolution algorithms - Burrus - Unscrambling for fast DFT algorithms - Burrus - A connection between bitreverse and matrix transpose.
Hardware and software consequences - Duhamel, Prado.
Rephrasing the convolution theorem, we get. Then we multiply this signal with the other signal and calculate the integral of the overlapping part. An alternative approach has been suggested in [KoPa] , using the Good—Thomas prime-factor fast Fourier transform to decompose the global computation into smaller Fourier transform computations, implemented by the Winograd small fast Fourier transform algorithm and reducing some of the additions at the cost of some multiplications. Discrete Fourier transforms when the number of data samples is prime - Rader - When frequency domain multiplication is carried out in rectangular form there are cross terms between the real and imaginary parts. Focus on understanding multiplication using polar notation , and the idea of cosine waves passing through the system.
Digital filtering using polynomial transforms - Nussbaumer - Fast computational algorithms for bit reversal - Polge, Bhagavan, et al. These follow directly from the fact that the DFT can be represented as a matrix multiplication. If f is circularly shifted by m i. Note zero phase at frequencies -3 and 3, corresponding to cosine function input.
Also, remember that phases of components not really there i. For two length-N sequences x and y, the circular convolution of x and y can be written as. Thus x[-1] is the same as x[N-1].
This is just like regular convolution of the same input sequences, except that it returns a vector of the same length as the two inputs, and it assumes periodicity to get values "off the edge", rather than assuming zero values. If we write circular convolution of x and y as x y, and element-wise multiplication of x and y in Matlab fashion as x.
Circular shift of input If f is circularly shifted by m i. Figure d shows three periods of how the DFT views the output signal in this example. Frequency domain convolution tries to place the point correct output signal , shown in c , into each of these point periods.
This results in 49 of the samples being pushed into the neighboring period to the right, where they overlap with the samples that are legitimately there. These overlapping sections add, resulting in each of the periods appearing as shown in e , the circular convolution. Once the nature of circular convolution is understood, it is quite easy to avoid.
For example, the signals in a and b could be padded with zeros to make them points long, allowing the use of point DFTs. After the frequency domain convolution, the output signal would consist of nonzero samples, plus samples with a value of zero. Chapter 18 explains this procedure in detail. Why is it called circular convolution?
Look back at Fig. Since all of the periods are the same, the portion of the signal that flows out of this period to the right , is the same that flows into this period from the left.
If you only consider a single period, such as in e , it appears that the right side of the signal is somehow connected to the left side. Imagine a snake biting its own tail; sample is located next to sample 0, just as sample is located next to sample When a portion of the signal exits to the right, it magically reappears on the left. In other words, the N point time domain behaves as if it were circular.
In the last chapter we posed the question: does it really matter if the DFT's time domain is viewed as being N points, rather than an infinitely long periodic signal of period N? Circular convolution is an example where it does matter.
If the time domain signal is understood to be periodic , the distortion encountered in circular convolution can be simply explained as the signal expanding from one period to the next. In comparison, a rather bizarre conclusion is reached if only N points of the time domain are considered. That is, frequency domain convolution acts as if the time domain is somehow wrapping into a circular ring with sample 0 being positioned next to sample N Smith, Ph.
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