WO2003079219A1 - Procede et dispositif visant a reduire la complexite de calcul de la transformee de fourier rapide d'un signal contenant des tonalites - Google Patents

Procede et dispositif visant a reduire la complexite de calcul de la transformee de fourier rapide d'un signal contenant des tonalites Download PDF

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Publication number
WO2003079219A1
WO2003079219A1 PCT/CA2003/000019 CA0300019W WO03079219A1 WO 2003079219 A1 WO2003079219 A1 WO 2003079219A1 CA 0300019 W CA0300019 W CA 0300019W WO 03079219 A1 WO03079219 A1 WO 03079219A1
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WO
WIPO (PCT)
Prior art keywords
data points
domain samples
radix
tones
frequency
Prior art date
Application number
PCT/CA2003/000019
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English (en)
Inventor
Dongxing Jin
Ping Wai Wan
Derrick Remedios
Leonard Marziliano
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Tropic Networks Inc.
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Publication date
Priority claimed from CA002377623A external-priority patent/CA2377623C/fr
Application filed by Tropic Networks Inc. filed Critical Tropic Networks Inc.
Priority to AU2003201555A priority Critical patent/AU2003201555A1/en
Publication of WO2003079219A1 publication Critical patent/WO2003079219A1/fr

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Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/147—Discrete orthonormal transforms, e.g. discrete cosine transform, discrete sine transform, and variations therefrom, e.g. modified discrete cosine transform, integer transforms approximating the discrete cosine transform
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/141—Discrete Fourier transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/141—Discrete Fourier transforms
    • G06F17/142—Fast Fourier transforms, e.g. using a Cooley-Tukey type algorithm

Definitions

  • a WDM- (wavelength- division multiplexed) optical signal carrying a plurality of channels has impressed upon each of its channels a 0 respective unique dither resulting in each channel having a unique tone.
  • the channels are modulated via amplitude modulation resulting in AM (Amplitude Modulation) tones each having a fixed modulation depth, for example, of approximately 8%.
  • AM Amplitude Modulation
  • Other modulation schemes are also used.5 Since the tones have a fixed modulation depth, channel power is a function of the tone power and channel power is measured by detecting the tones ' of fixed modulation depth.
  • N time domain samples of the power of the WDM0 optical signal are collected at a, sampling frequency, f s .
  • N frequency domain samples from N time domain samples
  • DFTs require a number of arithmetic operations of the order of N 2 .
  • a- conventional radix-M FFT requires on the order of Nlog M (N) arithmetic0 operations. FFTs are therefore computationally efficient when compared to DFTs even for N as low as 100.
  • a conventional radix-M FFT requires that the N frequency domain samples be computed simultaneously. Generally, only a fraction of the N frequency domain samples contain tones and as such only those frequency domain samples containing tones are required. Therefore since a portion, which can be significant, of the N frequency domain samples calculated are not required, the efficiency of the conventional radix-M FFT is compromised.
  • Various methods and apparatuses are provided for performing a radix-M FFT (Fast Fourier Transform) upon N time domain samples to produce N/S frequency domain samples for detecting tones of dithers impressed on channels of a WDM (wavelength Division Multiplexed) optical signal.
  • radix-M FFT Fast Fourier Transform
  • the methods and apparatuses may be used to measure channel power.
  • the radix-M FFT may be used to operate on a sequence of 2N real valued time domain samples by re- arranging the 2N real valued time domain samples into a
  • N/M r radix-M computations may be performed upon its respective subset of the N data points. Furthermore, for a stage, r, of the k stages wherein w ⁇ r ⁇ k, N/M +1 radix-M computations may be performed upon its respective subset of the N data points.
  • the method may be applied to a WDM (Wavelength. Division Multiplexed) optical signal having a plurality of channels. Some of the channels may each have impressed upon itself a unique dither resulting in a respective unique tone.
  • the unique tones may be detected and then converted into a power.
  • a method of performing a radix-M FFT where M is an integer satisfying M>2.
  • the method includes sampling a signal, containing tones, with a sampling frequency, f s , to produce a sequence of 2N real valued time domain samples, wherein N is an integer.
  • the sequence of 2N real valued time domain samples is split into two sequences of N real valued data points and the two sequences of N real valued data points are combined into a sequence of N complex valued data points.
  • a processing apparatus used to perform a radix-M FFT upon a sequence of 2N real valued time domain samples, wherein N and M are integers with M>2.
  • the 2N real valued time domain samples are sampled at a. sampling frequency, f s , from a signal containing tones to produce frequency domain samples that contain the tones.
  • the apparatus has a memory which is used to store data comprising the sequence of 2N real valued time domain samples.
  • the apparatus also has a processor capable of accessing the memory. The processor is used to split the sequence of 2N real valued time domain samples into two sequences of N real valued data points and combine the two sequences of N real valued data points into a sequence of N complex valued data points.
  • the respective subset contains only data points upon which the frequency omain samples that contain the tones are dependent.
  • the processor then applies a split function only to data points of the sequence of N complex valued data points upon which the frequency domain samples that contain the tones are dependent.
  • the sampling frequency, f s is such that the frequency domain samples contain the tones.
  • an article of manufacture has a computer usable medium having computer readable program code means embodied therein for causing a radix-M FFT upon a sequence of N time domain samples, wherein N and M are integers with M>2.
  • the .N time domain samples are sampled at a sampling frequency, f s , from a signal containing tones to produce frequency domain samples that contain the tones.
  • the N time domain samples each initialize a respective one of N data points.
  • the respective subset contains only data points upon which the frequency domain samples that contain the tones are dependent.
  • the article has computer readable code means for applying a split function only to data points of the sequence of N complex valued data points upon which the frequency domain samples are dependent. This is done after the radix-M computations. are performed for each- 'one of k stages.
  • Figure IB is a diagram of a conventional 4-stage radix-2 FFT (Fast Fourier Transform) using DIF (Decimation In Frequency) ;
  • Figure ID is a diagram of bit reversal operations of the conventional 4-stage radix-2 FFT of Figure IB;
  • Figure 3B is a diagram of a 4-stage radix-2 FFT using DIF, provided by another embodiment of the invention.
  • Figure 3C is a diagram of bit reversal operations of the 4-stage radix-2 FFT of Figure 3B;
  • Figure 5A is a diagram ' of N frequency domain samples of frequency bandwidth, ⁇ f, showing four frequency domain samples of interest each containing a respective one of four tones that require detection;
  • Figure 5B is a flow chart of a method used to perform a k-stage radix-M FFT upon N time domain samples associated with the N frequency domain samples of Figure 5A, provided by another embodiment of the invention;
  • Figure 7B is a table showing the correspondence between least significant bits of an index n and least T A03/00019
  • a WDM (wavelength- division multiplexed) optical signal carrying a plurality of channels ' has impressed upon at least one of its channels a unique dither resulting in each channel having a unique tone.
  • the channels are modulated via amplitude modulation resulting in AM (Amplitude Modulation) tones each having a fixed modulation depth, for example, of approximately 8%.
  • AM Amplitude Modulation
  • Other modulation schemes are also used. Since the tones have a fixed modulation depth, channel power is a function of the tone power and channel power is measured by detecting the tones of fixed modulation depth.
  • N time domain samples of the power of the WDM optical signal are collected at a sampling frequency, f s .
  • a DFT Discrete Fourier
  • a FFT Fast Fourier Transform
  • N frequency domain samples from N time domain samples
  • DFTs require a number of arithmetic operations of the order of N 2 ;
  • FFTs are therefore computationally efficient when compared to DFTs even for N as low as 100.
  • a conventional FFT requires that the N frequency domain samples be computed simultaneously. Generally, only a fraction of the N frequency domain samples contain tones and as such only those frequency domain samples containing tones are of interest and required. Therefore since only a portion of the N frequency domain samples calculated are of interest and required, the efficiency of the FFT is compromised.
  • a set of 8 radix-2 computations 105 are performed and new values for the data points X(i) result at 110.
  • a radix-2 computation a computation is performed for each one of two points.
  • a radix- 2 computation is performed at 155 on two data points X(b) and X ⁇ £ ) (in the first stage 101, 0 ⁇ b ⁇ 7 and 8 ⁇ 15) at 160 and 165, respectively, and new values of X(b) and X ( £ ) result at 170 and 175, respectively.
  • the sampling frequency, f s is also less than or equal to a maximum sampling frequency, f s , ma ⁇ / at which the time domain sample can be sampled.
  • the maximum sampling frequency, f s , ma ⁇ may be due to, for example, limitations on hardware used to collect the time domain samples.
  • the tones 150 are shown in Figure 2B which shows a diagram of a 4-stage radix-2 FFT computation using DIF, provided by an embodiment of the invention.
  • the tones 150 are contained in the frequency domain samples 200, 210, 220 which correspond to data points X(4), X(8), X(12), respectively.
  • the data points X(4), X(8), X(12) are of interest, however, as discussed above with reference to Figure IB, in a FFT the data points X(i) must be re-ordered.
  • the data points X(l), X(2), X(3) at 260 are mapped onto the data points X(4), X(8), X(12) at 250.
  • a line 185 separates radix-2 computations of the sets of 8 radix-2 computations 105, 115, 125, 135 which are calculated from those which are not calculated. More particularly, in the first stage 101, eight radix-2 computations are performed for the set of 8 radix-2 computations 105. In the second stage 102 only four radix-2 computations are performed for a sub-set of 4 radix-2 computations 270 of the set of 8 radix-2 computations 115. In the third stage 103 two radix-2 computations are performed for a sub-set of 2 radix-2 computations 280 of the set of 8 radix-2 computations 125.
  • Bit reversal operations 112 are used to map the data points X(p) at 215 onto the data points X(q) at 225.
  • Figure 2D also shows an offset, from the center frequencies 214, 224, 234, of the tone frequencies, f ta , of the tones 150 at 212, 222, 232, respectively.
  • frequency leakage may affect the accuracy of results obtained from a FFT.
  • FIG. 3B shown is a diagram of a 4- stage radix-2 FFT using DIF, provided by another embodiment of the invention.
  • the two tones 380 are shown with corresponding data points X(2) and X(10) .
  • lines 302 and 312 separate the radix-2 computations which are calculated from those which are not calculated. More particularly, in the first stage 101 eight radix-2 computations are performed for the set of 8 radix-2 computations 105. In the second stage 102, only four radix- 2 computations are performed for the sub-set of 4 radix-2 computations 270 of the set of 8 radix-2 computations 115.
  • the third stage 103 only two radix-2 computations are performed for a sub-set of 2 radix-2 computations 375 of the set of 8 radix-2 computations 125.
  • the fourth stage 104 only one radix-2 computation is performed for a sub-set of 1 radix-2 computation 385 of the set of 8 radix-2 computations 135.
  • the data points X(4), X(5) are mapped onto X(2), X(10), .respectively using bit reversal operations.
  • Figure 3C shows bit reversal operations 112 for all data points, X(i), in the example, only the two data points X(4), X(5) of the • block of data 320 at 215 are re-ordered using a 1-bit bit . reversal operation.
  • a radix-M FFT is performed using DIT (Decimation In Time) .
  • Shown in Figure 4A is a diagram of a 4-stage FFT using DIT, provided by another embodiment of the invention.
  • the radix-2 computations of the sets of 8 radix-2 computations 105, 115, 125, 135 are computed using a corresponding equation for DIT.
  • Figure 4B shows a diagram of a radix-2 computation of the sets of radix-2 computations 105, 115, 125, 135, of Figure 4A.
  • a radix-2 computation is performed at 455 on two data points X(b) and X ⁇ £ ) at 460 and 465, respectively, to obtain new values for X(b) and X ⁇ £ ) at 470 and 475, respectively.
  • R t (where R is an integer with R t ⁇ l) dithers are impressed on a signal having Q (where Q is an integer with Q ⁇ l) channels.
  • Q where Q is an integer with Q ⁇ l
  • the sampling frequency, f s is also less than or equal to the maximum sampling frequency, f s , max , at which the time domain sample can be sampled.
  • the maximum sampling frequency, f s ,max/ may be due to, for example, limitations on hardware used to collect the time domain samples.
  • f s , ma ⁇ may be 250 KHz.
  • ⁇ f a - S ⁇ f
  • step 5B-3 For each stage, r, (step 5B-3) , wherein l ⁇ r ⁇ w, N/M r radix-M computations are performed (step 5B-4) . More particularly, the N/M r radix-M computations are performed upon a respective subset of the N data points, upon which the data points of interest, X(n'+aS) , are dependent.
  • step 5B-5 if all stages, r, wherein l ⁇ r ⁇ w have been done then go to step 5B-6; otherwise return to step 5B-3.
  • step 5B-6 for each stage, r, wherein w ⁇ r ⁇ k, N/M w+1 radix-M computations are performed (step 5B-7) .
  • step 5B-8 if all stages, r, wherein 3 ⁇ r ⁇ 17 have been done then data points within the block of data of block number, d, are re-ordered using bit reversal operations (step 5B-9) ; otherwise return to step 5B-6.
  • N 16K
  • Ncop 1.75N
  • N 16K
  • FIG. 6A shown is a block diagram of a processing platform used to perform the k-stage radix-M FFT of Figure 5B.
  • the processing platform generally indicated by 10, includes a CPU (Central Processor Unit) 12, an internal memory 14, a bus 16, an external memory 18, and a DMA (Direct Memory Access) unit 20.
  • the internal memory 14 is directly accessible by the CPU 12 and when the CPU 12 needs to operate on data stored in the internal memory 14, the CPU 12 retrieves the data directly from the internal memory 14.
  • the external memory 18 is indirectly accessible by the CPU 12 through the DMA unit 20.
  • the CPU 12 issues a command to the DMA unit 20 to retrieve the data.
  • the DMA unit 20 accesses the data in the external memory .18, and imports it across the bus 16 into the internal memory 14, where the processor 12 can process the data.
  • the internal -memory 14 does not have enough memory to store both the data being retrieved and existing data residing in internal memory 14, memory must be freed up. To do so the DMA unit 20 accesses the existing data in the internal memory 14, and exports it across the bus 16 into the external memory 18 before any data is moved into the internal memory 14.
  • the processor 12 may generate output data. When the CPU 12 is finished processing the data, the CPU 12 passes the output data back into the internal memory 14.
  • Figure 6A illustrates only the memory used for performing the k-stage radix-M FFT.
  • the internal memory is larger than illustrated; the processing platform 10 may perform in parallel other operations.
  • the DMA unit 20 can work independently from the CPU 12 and therefore, while the CPU 12 accesses one memory location the DMA unit 20 can access another memory location, so that full use of the processing platform 10 'capabilities is achieved in this way.
  • Bytes 8 Bytes of memory for storage.
  • the radix-2 computations each require two data points of 8 Bytes and one twiddle factors of 8 Bytes for a total of 24 Bytes for each radix-2 computation.
  • N/M r radix-M computations are performed.
  • N 128K
  • Figure 6B shows a flow chart of a method used to perform the radix-M computations of the stages, r, where l ⁇ r ⁇ k, of k stages of the k-stage ' radix-M FFT of Figure 5B (steps 5B-3 to 5B-8) . More particularly, Figure 6B shows a flow chart of a method used to perform steps 5B-3 to 5B-8 of Figure 5B .
  • M 2 in the illustrative example
  • the CPU 12 performs M R radix- M computations upon the data points in the internal memory 14 (step 6B-3) .
  • the DMA unit 20 then exports the data points from the internal memory 14 back into the external memory 18 (step 6B-4) .
  • step 6B-5 if radix-M computations have been performed for all N t time steps then the steps are finished; otherwise return to step 6B-1.
  • N 128K
  • the DMA unit 20 must import and export data points and respective twiddle factors (steps 6B-2 and 6B-4)
  • a k * 12 bit index p
  • a k" 12 bit index q, wherein q is the bit reversal of p, is required to be stored in a look-up table in the internal memory 14.
  • n 0, 1, ...,N-1 and R(n) and I(n) real and imaginary parts of Y(n), respectively.
  • the real and imaginary parts X' r (n) and X-(n) are given by
  • R t (where R t is an integer with , R t ⁇ l) dithers are impressed on a signal having Q (where Q is an integer with Q ⁇ l) channels.
  • FIG. 7A shown is a flow chart of a method used to obtain the frequency domain samples X'(n'+aS) from the sequence of 2N real valued time domain samples, x(c), provided by another embodiment of the invention.
  • Corresponding w 2 least significant bits of the index N-n are given at 780.

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Abstract

L'invention concerne divers procédés et dispositifs permettant de mettre en oeuvre une transformée de Fourier rapide (TFR) de base M sur N échantillons de domaine temporel afin de produire N/S échantillons de domaine fréquentiel permettant de détecter des tonalités de vibration imprimées à des canaux d'un signal optique multiplexé en longueur d'onde (WDM). Des tonalités successives comportent un espacement de fréquences de tonalités, ?fta et une fréquence d'échantillonnage, fs, est choisie de sorte que fs = N?fta/S. S représente un espacement donné par S = Mw, w représentant un nombre entier. La TFR de base M est mise en oeuvre en k = logM(N) étapes ; et dans ces étapes, un nombre réduit de calculs de base M, par rapport au nombre de calculs de base M d'une TFR de base M classique, sont effectués sur des points de données associés aux N échantillons de domaine temporel. Ces opérations sont possibles car les échantillons successifs de domaine fréquentiel des N/S échantillons de domaine fréquentiel diffèrent selon ?fta = S?f, ?f représentant une largeur de bande de fréquences.
PCT/CA2003/000019 2002-03-20 2003-01-10 Procede et dispositif visant a reduire la complexite de calcul de la transformee de fourier rapide d'un signal contenant des tonalites WO2003079219A1 (fr)

Priority Applications (1)

Application Number Priority Date Filing Date Title
AU2003201555A AU2003201555A1 (en) 2002-03-20 2003-01-10 Method and apparatus to reduce the comlexity of the calculation of the fast fourier transform of a signal containing tones

Applications Claiming Priority (4)

Application Number Priority Date Filing Date Title
CA2,377,623 2002-03-20
CA002377623A CA2377623C (fr) 2002-03-20 2002-03-20 Methode et appareil de reduction par calcul en vue de la detection de tonalites
US10/134,382 US6732058B2 (en) 2002-03-20 2002-04-30 Method and apparatus for computation reduction for tone detection
US10/134,382 2002-04-30

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Citations (3)

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Publication number Priority date Publication date Assignee Title
US5365469A (en) * 1990-10-31 1994-11-15 International Business Machines Corporation Fast fourier transform using balanced coefficients
EP0809194A2 (fr) * 1996-03-28 1997-11-26 Simmonds Precision Products Inc. Conditionneur universel de signaux à bande étroite
US5774388A (en) * 1993-08-11 1998-06-30 France Telecom Device for electronically calculating a fourier transform and method of minimizing the size of internal data paths within such a device

Patent Citations (4)

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Publication number Priority date Publication date Assignee Title
US5365469A (en) * 1990-10-31 1994-11-15 International Business Machines Corporation Fast fourier transform using balanced coefficients
US5774388A (en) * 1993-08-11 1998-06-30 France Telecom Device for electronically calculating a fourier transform and method of minimizing the size of internal data paths within such a device
EP0809194A2 (fr) * 1996-03-28 1997-11-26 Simmonds Precision Products Inc. Conditionneur universel de signaux à bande étroite
US5912829A (en) * 1996-03-28 1999-06-15 Simmonds Precision Products, Inc. Universal narrow band signal conditioner

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Title
EXPOSITO A G ET AL: "FAST NON-RECURSIVE COMPUTATION OF INDIVIDUAL RUNNING HARMONICS", IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, IEEE INC. NEW YORK, US, vol. 47, no. 8, August 2000 (2000-08-01), pages 779 - 782, XP001075100, ISSN: 1057-7130 *
LYONS R: "WINDOWING FUNCTIONS IMPROVE FFT RESULTS", TEST AND MEASUREMENT WORLD. (INC. ELECTRONICS TEST ), CAHNERS PUBLISHING, DENVER, US, vol. 18, no. 10, 1 September 1998 (1998-09-01), pages 53 - 54,56,58,60, XP000779830, ISSN: 0744-1657 *
MITRA S K ET AL: "DFT CALCULATION VIA SUBBAND DECOMPOSITION", SIGNAL PROCESSING 5: THEORIES AND APPLICATIONS. PROCEEDINGS OF EUSIPCO-90 FIFTH EUROPEAN SIGNAL PROCESSING CONFERENCE. BARCELONA, SEPT. 18 - 21, 1990, PROCEEDINGS OF THE EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO), AMSTERDAM, ELSEVIER, NL, vol. 1 CONF. 5, 18 September 1990 (1990-09-18), pages 501 - 504, XP000358165, ISBN: 0-444-88636-2 *

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