Patent Application
20190332642
The present invention relates generally to the calculation of Fourier transforms and more specifically to a system and method for quickly calculating Fourier transforms without multiplication.
The discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The DFT has widespread use in many areas of science and engineering, including biomedical engineering, telecommunications and astronomy.
Oscillating functions in the form of a scalar multiplied by a complex exponential are an essential building block in computational mathematics, and particularly in the DFT. Such functions take the form:
ƒ(t)=Aeiωt (Equation 1)
The DFT takes the form:
where N is the number of samples. Direct calculation of the DFT according to Equation 2 can be difficult, as it involves O(N2) (order of N2) multiplications, and thus becomes prohibitively complex and time consuming for large values of N.
As is known in the art, one known technique to mitigate this difficulty is the algorithm to reduce the complexity of the DFT calculations that is known as the fast Fourier transform (FFT). The FFT factors the DFT matrix into a product of sparse (mostly zero) factors, and uses recursive application of a “butterfly” unit to combine the results of smaller transforms into a larger transform. The FFT can reduce the multiplicative complexity of the transform from O(N2) to O(N log(N)), a substantial improvement over the direct DFT calculation. While the difference in speed can be great, particularly when N is in the thousands or millions, the FFT may remain computationally demanding, often still involving hundreds or thousands of multiplications.
Further, the FFT does not address the need for generating the roots of unity in Equation 2; the roots of unity are used to implement the angular rotation of the transform and give rise to the bulk of the complexity in the conventional implementation of the FFT due to the need for complex multiplications between these numbers and the input samples. This requires tabulated storage of these so-called “twiddle factors” in Read-Only Memory (ROM). As a consequence, microelectronic implementations of the FFT require many clock cycles to complete a single transform and occupy a large chip area.
To avoid these issues, it would be useful to be able to calculate Fourier transforms quickly and without the need to perform complex multiplications.
An improved system and method for quickly calculating Fourier transforms without multiplication is disclosed.
One embodiment discloses a circuit for calculating a quadrature output having X and Y values from a quadrature input having X and Y values which define a point on a sinusoidal waveform, comprising: a first scaling element configured to receive the input Y value and scale the input Y value by a scaling factor; a first adder configured to receive and add the input X value and the scaled Y value thereby producing the output X value; a second scaling element configured to receive the output X value and scale the output X value by the scaling factor; and a second adder configured to receive and add the input Y value and the scaled output X value thereby producing the output Y value; wherein the output X value and output Y value define a next point on the sinusoidal waveform.
Another embodiment discloses a circuit for performing a Fourier transform on samples of a signal, comprising: a chain of N combinatorial units, where N is a number of samples of a sine wave to be included in the Fourier transform, each combinatorial unit comprising: a first scaling element configured to receive an input Y value and scaling the Y value by a scaling factor; a first adder configured to receive and add an input X value and the scaled Y value thereby producing an output X value; a second scaling element configured to receive the output X value and scale the X value by the scaling factor; and a second adder configured to receive and add the input Y value and the scaled output X value thereby producing an output Y value; a first combinatorial unit in the chain receiving an input signal, and each subsequent combinatorial unit receiving as its input the output X and Y values of an immediately preceding combinatorial unit in the chain; a stateful adder configured to add selected combinations of outputs from the combinatorial units thereby determining a current transform element and add the current transform element to a prior transform sum to determine a current transform sum; and an output element configured to store the current transform sum and output the current transform sum after the series of quadrature inputs is complete.
Still another embodiment discloses a method for performing a Fourier transform on a signal, comprising: receiving by a circuit a series of quadrature inputs representing an input signal, each quadrature input having X and Y values; for each quadrature input, creating by the circuit a plurality of samples of a sine wave to be included in the Fourier transform by repeatedly: scaling by the circuit an input Y value by a scaling factor; adding by the circuit an input X value and the scaled Y value thereby producing an output X value; scaling by the circuit the output X value by the scaling factor; and adding by the circuit the input Y value and the scaled output X value thereby producing an output Y value; for each quadrature input, adding by the circuit selected combinations of sine wave samples thereby determining a current transform element; and adding by the circuit all of the current transform elements from each quadrature input.
Yet another embodiment discloses a method for calculating a quadrature output having X and Y values from a quadrature input having X and Y values which define a point on a sinusoidal waveform, comprising: scaling by the circuit the input Y value by a scaling factor; adding by the circuit the input X value and the scaled Y value thereby producing an output X value; scaling by the circuit the output X value by the scaling factor; and adding by the circuit the input Y value and the scaled output X value thereby producing an output Y value; wherein the output X value and output Y value define a next point on the sinusoidal waveform.
Another embodiment discloses a circuit for performing a discrete Fourier transform on four samples of a signal, comprising: first and second arithmetic units each configured to receive a different two of the samples as complex quadrature inputs having X and Y values, and each comprising first and second complex adders, the first complex adder adding the input X and Y values of the two samples, and the second complex adder adding the X and Y value of one sample to the inverse of the X and Y values of the other sample, thereby resulting in two quadrature outputs having X and Y components; first, second, third and fourth combinatorial units, each combinatorial unit comprising: a first scaling element configured to receive an input Y value and scaling the Y value by a scaling factor; a first adder configured to receive and add an input X value and the scaled Y value thereby producing an output X value; a second scaling element configured to receive the output X value and scale the X value by the scaling factor; and a second adder configured to receive and add the input Y value and the scaled output X value thereby producing an output Y value; the first and third combinatorial units configured to receive a first quadrature output of the first arithmetic unit, and the second and fourth combinatorial unit configured to receive a second quadrature of the first FFT unit; first, second, third and fourth adders; the first adder configured to add the first output of the second arithmetic unit to the output of the first combinatorial unit; the second adder configured to add the second output of the second arithmetic unit to the output of the second combinatorial unit; the third adder configured to add the first output of the second arithmetic unit to the output of the third combinatorial unit; and the fourth adder configured to add the Y output of the second arithmetic unit to the output of the fourth combinatorial unit.
An improved system and method is provided for calculating Fourier transforms using only combinatorial elements, thus completely eliminating the need for multiplication. A combinatorial unit circuit that contains only adders and scaling elements is used to calculate each value, as in Equation 1 above, that is then summed to create the Fourier transform as in Equation 2 above.
In various embodiments, the system and method utilize the fact that a real-time Fourier transform (RTFT) that includes all the multiplications and roots of unity in Equation 2 can be implemented through only a number of such combinatorial units, making it possible to complete a hardware DFT transform in a single clock cycle regardless of the value of N, and eliminating the need for a twiddle factor ROM used in the FFT. This brings a significant improvement in speed and simplicity over the prior art.
It is known that the exponential value in Equation 1 can be written as:
e
iωt=cos ωt+i sin ωt (Equation 3)
which describes a circle in a plane having a real axis and an orthogonal imaginary axis.
In 1972, Marvin Minsky of MIT, in an exploration of early computer graphics, noticed that a simple algorithm of the form:
X
1
=X
0
−εY
0
Y
1
=Y
0
−εX
1 (Equation 4)
produces a sequence of points (X,Y) that also forms a circle, Minsky M, MIT AI Memo 239, item 149, February 1972, and thus also a sine wave. The circles were noted to be imperfect, and the algorithm was thus not pursued in the computer graphics field. Despite the perceived imperfection, the circular response of Equation 4 is similar enough to that of Equation 1 (or the equivalent Equation 3) so as to represent a remarkably simple approach to generating a quadrature harmonic oscillation.
In fact, the imperfection of Equation 4 is due to a phase shift of half a step, and thus Equation 4 can be made to produce a mathematically perfect quadrature oscillator, in effect generating a sequence of roots of unity with every iteration or step. Further, although presented as a sequence of two assignments as in Equation 4, the algorithm has no state. This is essential to its function, as introducing a temporary state will cause the algorithm to diverge. This lack of state allows the algorithm to be represented as a unit of combinatorial logic, shown in
Any method or circuit known in the art can be used to implement scaling elements 110 and 130 and adders 120 and 140. In one embodiment the scaling elements are implemented as bit shifts, allowing data to flow through circuit 100 without the need of a clock, resulting in a near-immediate quadrature output (X′,Y′).
If the initial values X0 and Y0 are 1 and 0, respectively, Equation 4 will produce the values of a sine wave with an amplitude of 1. By changing the initial value of X to be an actual input value, Equation 4 results in the values of a sine wave that varies in amplitude based upon the initial value of X. (The initial value of Y typically remains 0, except in the case of a complex DFT.) Thus, each sample results in a string of all of the phase shifted values of the value of X as if the value of X has been multiplied by all of the points on a sine wave. These points are then summed in appropriate combinations to get the Fourier transform of X.
Multiple instances of combinatorial circuit 100 can thus be used to obtain all of the values of Equation 4, and thus of Equation 1, for a given input value X. These values can then be added in the desired combinations, as in Equation 2, to obtain the Fourier transform of X. In one embodiment, this allows all of the multiplications and unity roots of the DFT to be implemented instead in a purely combinatorial fashion as a simple chain of such circuits.
The function values from combinatorial circuits 220 are input to an adder 230, which adds selected values from the combinatorial circuits 220 to a sum stored in output element 240 so as to incrementally determine the DFT sum according to Equation 2 above; adder 230 may be considered a “stateful” adder in that it adds each received sum to the prior total stored in output element 240, which accumulates the final result over all of the input samples.
The stateful adder implements a sum of selected values from the combinatorial circuits 220 with a rolling “stride”, skipping over values with a step that is defined by the input sample position according to Equation 2. One of skill in the art will be able to determine the logic appropriate for stateful adder 230 to add the desired values to obtain each value to be added to the DFT sum. Output element 240 may be double buffered, so that when the last input value has been clocked in and the DFT calculation is complete, the transform result can be output on a clock cycle while a new input block is being input and processed on the same clock cycle.
Circuit 200 thus implements a fully streaming, multiplier-free DFT; the circuit 200 has zero latency beyond the time it takes to clock data into the chain 210 of combinatorial units 220. Circuit 200 can be implemented with a total of 3N adders, and no multipliers; as above, each of the N combinatorial units 220 has two adders, while stateful adder 230 has N individual adders, one for each of the N values to be added to calculate the transform according to Equation 2 (each of the adders in stateful adder 230 will have multiple inputs). Stateful adder 230 will also have the logic needed to determine which samples are to be added together as in Equation 2.
In some embodiments, adder 230 will have an equal number of input and output ports, while in other embodiments adder 230 will perform interpolation between the harmonic function elements in order to generate a smaller desired final transform size and/or improve noise.
The circuits 100 and 200 of
It will be obvious to those of skill in the art how to implement circuits 100 and 200 in digital form.
Circuit 300 calculates new values of Xout and Yout from quadrature input Xin and Yin according to Equation 4 above (with one exception noted below). The resistance of resistors R3 and R4 is greater than that of the other resistors by the factor E, thus providing the scaling factor of Equation 4.
Amplifiers A1 and A2 in circuit 300 have a high open loop gain and amplify their input by a large negative number, so that the output may, for example, be characterized as:
AmpOut=−1e6·AmpIn (Equation 5)
Note that circuit 300 thus results in the values of Xout and Yout being the inverse of the values predicted by Equation 4 above. This must be taken into account when constructing a chain of such analog combinatorial circuits.
As in
As above, the output of each combinatorial circuit of the form of circuit 300 inverts from the sign of the input. In practice this is easily handled; for example, if a circuit of the form of circuit 400 is used as the chain 210 of combinatorial circuits 220 in
As will be clear to one of skill in the art, in some embodiments, the symmetry of the unit circle can be exploited to reduce the number of combinatorial circuits in a chain from the number needed to cover a full cycle of a sine wave, and instead, by applying the symmetry of the sine wave, use only the number of elements sufficient to cover one quarter of the full sine wave cycle. Further, some units may be operated in reverse, allowing a chain length of only ⅛th of the full sine wave cycle. This may prevent timing issues from arising as data flows through the combinatorial units at high values of N.
The system and method described herein can also provide improvements in streaming applications. When a conventional DFT is applied to a stream of data, the block nature of the transform generates artifacts in the form of spectral bleed in the transform output, known as “windowing.” This is undesirable, particularly in telecommunication applications, where it can degrade the bandwidth of a transmission relying on the DFT, such as in orthogonal frequency-division multiplexing (OFDM). It is possible to avoid these artifacts with another embodiment of the present system and method.
While in circuit 200 one set of samples is clocked in on each clock cycle to determine one element of a Fourier transform, circuit 500 must operate faster due to the need for an output signal for each band, or frequency, of the input signal. One set of samples for each band must be processed in order to generate a single output sample; the sets of samples are taken in to circuit 500 serially.
This may be accomplished either by operating circuit 500 at a faster rate, or by providing more than one instance of circuit 500 operating in parallel. Thus, if the input signal to circuit 500 has, for example, 32 bands, and only a single circuit 500 is used, the clock for adder/multiplexor 530 must run at 32 times the speed of the system that generates the input signal.
The sine wave values from combinatorial circuits 520 are input to N-to-M adder/multiplexer 530 that in turn outputs a series of sine wave functions of different frequency generated by variation in stride as the multiplexer selects different combinations of values from the combinatorial chain on every clock. The resulting signals are input to adder 540, to output a sum of modulated quadrature waveforms. The samples for one band of the input signal are processed on each clock cycle, and one output sample is thus created every 32 clock cycles.
The need for a faster clock speed may be significant. For example, WiFi applications typically have 64 bands and operate at a 50 megahertz (MHz) base band before modulation. Thus, the clock in a single circuit 500 for such an application would need to run at 50×64=3200 MHz, or 3.2 gigahertz (GHz). The required clock speed will obviously be reduced if multiple circuits 500 are used in parallel, but at the cost of additional area occupied (on a chip, for example) by each such parallel circuit. It is possible to even place 32 such circuits in parallel, each operating at the same speed as the system generating the input signal, but with a needed area 32 times as large as a single circuit.
The output of circuit 500 provides a continuous Fourier transform which can then be modulated for a carrier wave (not shown), and is truly streaming without the windowing artifacts present in the conventional DFT. The latency is zero, as a new valid sample is generated immediately upon every new input. This can benefit transmitter implementations in telecommunications by reducing noise and maximizing speed and bandwidth, and it only requires 2N addition operations to implement, requiring far less computation than the traditional DFT equivalent embodiment.
Again, a quadrature signal (X,Y), which is the received signal (after the carrier wave is demodulated down to the base band, not shown), enters a chain 610 of combinatorial units 620 as with the circuits above. The combinatorial circuits 620 demodulate the received signal by a frequency in the same way that combinatorial circuits 520 modulated the input signal in circuit 500; just as the input signal to circuit 500 had to be modulated for each frequency band, the received signal in circuit 600 must be demodulated by each frequency.
Thus, as with circuit 500, if only a single instance of circuit 600 is used, it again must be operated at a speed faster than that of the underlying signal generation system, as now a single input quadrature signal must be used to generate all of the bands of the original signal input into circuit 500. Alternatively, as above, multiple instances of circuit 600 may be used, decreasing the required speed of each circuit but increasing the size of the overall circuit.
The resulting signals from combinatorial units 620 are input to N-to-M multiplexer 630 to combine the different demodulated frequency components of the received signal, each frequency entering one of low pass filters 640 to add a frequency component to the reconstructed original signal in double buffer adder 650, from which the output is generated at the same rate as the input to the original signal-generating system.
One of skill in the art will appreciate that many different processing elements can be configured to implement the rotation of signals represented as complex pairs as described herein.
Circuits such as circuit 100 of
In order to start the method of Cooley-Tukey, in circuit 800 of
The outputs of elements 800a and 800b are as shown in
As is well known in the art, this concept can be applied recursively to create FFTs of larger sizes. Thus, the outputs of circuit 900 may be applied to two more instances of a circuit of configuration similar to circuit 900, but in which circuits 800a and 800b are replaced by instances of circuit 900 itself, and the rotations produced by combinatorial units 920 now result in eight rotations, i.e., every 45 degrees. This may be followed by four more such circuits to obtain 16 equally spaced rotations, etc., each circuit using the prior circuit in place of circuits 800a and 800b. In this way a circuit capable of performing a FFT of any desired size of N may be constructed.
The benefit of using the circuit 100 from
At step 1004, a series of sine wave values is created for each quadrature input. This is done sequentially, i.e., the sine wave samples are created for the first quadrature input, then the second quadrature input after the first quadrature input has been transformed as below, etc.
In steps 1006 to 1012, the sine wave samples are created according to Equation 4 above, for example, by combinatorial units such as circuit 100 of
At step 1010, the output X value is scaled by the scaling factor, and at step 1012 the input Y value and the scaled output X value are added to obtain an output Y value.
At step 1014, selected combinations of the sine wave values of the quadrature input are added to obtain a transform element of the Fourier transform, for example, by stateful adder 230 of
At step 1016, all of the transform elements from all of the quadrature inputs are added to obtain the Fourier transform of the quadrature input series.
It will be apparent that steps 1006 to 1012 comprise a method corresponding to circuit 100 of
In another embodiment the system and method described herein can be used to compute the Discrete Cosine Transform (DCT). The DCT has important applications in image, audio and video compression and is widely used in lossy compression encoders such as MPEG and JPEG. The N-point DCT can be obtained by performing a 4N- or 2N-point RTFT on a zero-padded or mirrored input vector, according to any of the known methods in the art for deriving the DCT from the DFT. The technique described herein provides one means to perform zero-latency compression for high speed real-time applications.
In still a further embodiment the described system and method can be used for finite Fourier series approximations of periodic functions such as sawtooth and triangle waveforms, by realizing fixed combinations of oscillating function terms with combinatorial logic. This provides new ways to efficiently generate complex waveforms for signal processing applications.
It should be appreciated that the described method and apparatus can be implemented in numerous ways, including as a process, an apparatus, or a system. The methods described herein may be implemented by program instructions for instructing a processor to perform such methods, and such instructions recorded on a non-transitory computer readable storage medium such as a hard disk drive, floppy disk, optical disc such as a compact disc (CD) or digital versatile disc (DVD), flash memory, etc. It may be possible to incorporate some methods into hard-wired logic if desired. It should be noted that the order of the steps of the methods described herein may be altered and still be within the scope of the disclosure.
It is to be understood that the examples given are for illustrative purposes only and may be extended to other implementations and embodiments with different conventions and techniques. While a number of embodiments are described, there is no intent to limit the disclosure to the embodiment(s) disclosed herein. On the contrary, the intent is to cover all alternatives, modifications, and equivalents apparent to those familiar with the art.
In the foregoing specification, the invention is described with reference to specific embodiments thereof, but those skilled in the art will recognize that the invention is not limited thereto. Various features and aspects of the above-described invention may be used individually or jointly. Further, the invention can be utilized in any number of environments and applications beyond those described herein without departing from the broader spirit and scope of the specification. The specification and drawings are, accordingly, to be regarded as illustrative rather than restrictive. It will be recognized that the terms “comprising,” “including,” and “having,” as used herein, are specifically intended to be read as open-ended terms of art.
This application claims priority to Provisional Application No. 62/662,542, filed Apr. 25, 2018, which is incorporated by reference herein in its entirety.
| Number | Date | Country | |
|---|---|---|---|
| 62662542 | Apr 2018 | US |
