CONTINUOUS FOURIER TRANSFORM METHOD AND APPARATUS

• US 3,851,162 A
• Filed: 04/18/1973
• Issued: 11/26/1974
• Est. Priority Date: 04/18/1973
• Status: Expired due to Term

• Current Assignee: Robert M. Munoz
• Original Assignee: Robert M. Munoz
• Inventors: Munoz, Robert M.
• Primary Examiner(s): Gruber, Felix D.
• Assistant Examiner(s): Gottman, James F.
• Application Number:
  US05/352,382
• Time in Patent Office: 587 Days
• Field of Search: 235/156,152,184,181,197
• US Class Current: 708/821
• CPC Class Codes: G06G 7/1921 for forming Fourier integrals, harmonic analysis and synthesis spectrum analysis G01R23/00
• CPC Class Codes: H04B 1/66 for reducing bandwidth of signals; for improving efficiency of transmission H04B1/68 takes precedence
• CPC Class Codes: H04B 3/141 using multiequalisers, e.g. bump, cosine, Bode

Abstract

An input analog signal to be frequency analyzed is separated into N number of simultaneous analog signal components each identical to the original but delayed relative to the original by a successively larger time delay. The separated and delayed analog components are combined together in a suitable number of adders and attenuators in accordance with at least one component product of the continuous Fourier transform and analog signal matrices to separate the analog input signal into at least one of its continuous analog frequency components of bandwidth 1/N times the bandwidth of the original input signal. Given the separated frequency components, the original analog input signal can be reconstituted by combining the separate analog frequency components in accordance with the component products of the continuous Fourier transform and analog frequency component matrices. The continuous Fourier transformation is useful for spectrum analysis, filtering, transfer function synthesis, and communications.

Claims

1. In a method for continuous analog analysis for a time varying input function f(t) in the time domain into at least one of its Fourier Frequency components Cm(t) the steps of:

continuously producing N signal components fm(t)/N ranging from f0(t)/N to fN 1(t)/N wherein each signal component except f0(t)/N is delayed an amount T from the previous signal component;

continuously combining separated ones of said signal componeNts (fm(t)) fm(t)/N together so as to produce at least one continuous output function Cm(t) corresponding to one of the Fourier frequency components of the time varying input function f(t) to be frequency analyzed.

2. The method of claim 1 wherein the step of combining said separated components fm(t)/N together so as to produce at least one output function Cm(t) corresponding to one of the Fourier frequency components of the time varying input signal f(t) comprises the step of, combining the time delayed continuous signal components fm(t)/N together according to at least one of the component products of the matrix product:

DF(t) where D is the discrete Fourier transform matrix and F(t) is the delayed input signal component matrix.

3. The method of claim 2 wherein the step of combining the delayed signal components fm(t)/N according to at least one component product of the matrix product includes the step of, adding the delayed signal components fm(t)/N according to at least one of the component products of said matrix product.

4. The method of claim 2 wherein D is defined by the matrix:

5. The method of claim 1 wherein N is a power of 2.

6. The method for reconstituting a time varying function f(t) in the time domain from a plurality of N number of its continuously time varying Cm(t) Fourier frequency components in the frequency domain the steps of:

continuously combining the separate Fourier frequency components Cm(t) according to the product of the matrix equation f(t) DC(t) where D is the discrete Fourier transform matrix and C(t) is the matrix for the continuous separate Fourier frequency components.

7. The method of claim 6 wherein the step of combining a plurality of the continuous time varying Fourier frequency components Cm(t) together so as to reconstitute the original time varying function f(t) includes, combining the separate Fourier frequency components Cm(t) according to the product of the matrix Equation:

f(t) DC(t) where D is the discrete Fourier transform matrix and C(t) is the matrix for the continuous separate Fourier frequency components.

8. The method of claim 7 wherein the D matrix is defined by the matrix equation:

9. The method of claim 8 where

10. In an analog apparatus for continuous frequency analysis of a time varying input function f(t) in the time domain into at least one of its Fourier frequency components Cm(t):

means for continuously generating N signal components fm(t)/N ranging from f0(t)/N to fN 1(t)/N wherein each signaL component except f0(t)/N is delayed an amount T from the previous signal component;

means for continuously combining said signal components fm(t)/N together so as to produce at least one continuous output function Cm(t) corresponding to one of the Fourier frequency components of the time varying input function f(t) to be frequency analyzed.

11. The analog apparatus of claim 10 wherein the means for combining said separated signal components fm(t)/N together so as to produce at least one output function Cm(t) corresponding to one of the Fourier frequency components of the time varying input signal f(t) includes, means for combining the time delayed continuous signal components fm(t)/N according to at least one of the component products of the matrix product:

DF(t) where D is the discrete Fourier transform matrix and F(t) is the separated input signal component matrix.

12. The analog apparatus of claim 11 wherein said means for combining the delayed signal components according to at least one component product of the matrix product includes, means for adding the delayed signal components according to at least one of the component products of said matrix product.

13. The analog apparatus of claim 11 where D is defined by the matrix:

14. The analog apparatus of claim 10 wherein N is a power of 2.

15. Apparatus comprising:

first analog means for continuously producing N time varying Fourier frequency components Cm(t) in the frequency domain of a time varying function f(t) in the time domain;

said first analog means comprising;

means for continuously generating N signal components fm(t)/N ranging from f0(t)/N to fN 1(t)/N wherein each signal component except f0(t)/N is delayed an amount T from the previous signal component; and

means for continuously combining said signal components fm(t)/N together so as to produce N varying Fourier frequency components Cm(t);

channel means for conveying each time varying Fourier frequency component to an inverse Fourier transformer, said transformer comprising second analog means for combining the continuously time varying Fourier components Cm(t) together so as to reconstitute the original time varying function f(t).

16. The apparatus of claim 15 wherein said means for combining a plurality of the continuously time varying frequency components Cm(t) together so as to reconstitute the original time varying function f(t) includes, means for combining the separate frequency components Cm(t) according to the component products of the matrix product DC(t) where D is the discrete Fourier transform matrix and C(t) is the matrix for the continuous separate Fourier frequency components.

17. The apparatus of claim 16 wherein:

18. The apparatus of claim 17 wherEin:

19. In an analog apparatus for continuous frequency analysis of a time varying input function f(t) in the time domain into at least one of its Fourier frequency components Cm(t):

means for continuously generating N signal components fm(t)/N ranging from f0(t)/N to fN 1(t)/N wherein each signal component except f0(t)/N is delayed an amount T from the previous signal component;

said generating means including a tape recorder with a recording head and (N-2) staggered playback heads, said signal component f0(t)/N being connected to said recording head and each of the remaining signal components being produced at one of said playback heads;

means for continuously combining said signal components fm(t)/N together so as to produce at least one continuous output function Cm(t) corresponding to one of the Fourier frequency components of the time varying input function f(t) to be frequency analyzed.

20. An analog apparatus as set forth in claim 19 wherein said means for continuously combining signal components consists of a network of operational amplifiers.

21. Apparatus comprising:

first analog means for continuously producing N time varying Fourier frequency components Cm(t) in the frequency domain of a time varying function f(t) in the time domain;

said first analog means comprising;

means for continuously generating N signal components fm(t)/N ranging from f0(t)/N to fN 1(t)/N wherein each signal except f0(t)/N is delayed an amount T from the previous signal component; and

means for continuously combining said signal components fm(t)/N together so as to produce N varying Fourier frequency components Cm(t);

Said generating means including a tape recorder with a recording head and (N-2) staggered playback heads, said signal component f0(t)/N being connected to said recording head and each of the remaining signal components being produced at one of said playback heads;

channel means for conveying each time varying Fourier frequency component to an inverse Fourier transformer, said transformer comprising second analog means for combining the continuously time varying Fourier components Cm(t) together so as to reconstitute the original time varying function f(t);

said combining means consisting of a network of operational amplifiers.

22. Apparatus as set forth in claim 21 wherein said inverse Fourier transformer consists of a network of operational amplifiers and attenuators.