Frequency extension of harmonic signals
View Patent ↗A system and methods are provided for extending the frequency bandwidth of a harmonic signal. Harmonic content of a band-limited signal is extended to frequencies outside the signal's passband by performing a non-linear transformation on the complex spectrum of the band-limited signal in the frequency domain. The non-linear transformation may be accomplishes by a linear convolution of the complex spectrum with itself. A system for extending the frequency bandwidth of a harmonic signal includes a signal processor with a forward transform module for transforming a time domain signal into the frequency domain, a non-linear transform module for performing the non-linear transformation on the complex spectrum of the harmonic signal, and a reverse transform module for transforming the extended spectrum of the harmonic signal back into the time domain.
1. A computer-implemented method of extending harmonics of a band-limited harmonic signal through the use of a processor, the method comprising:
transforming, through the use of a processor, a band-limited harmonic signal from a time domain to a frequency domain to obtain a complex spectrum of the band-limited harmonic signal;
performing, through the use of the processor, a non-linear transformation on the complex spectrum of the band-limited harmonic signal in the frequency domain which extends harmonic content of the band-limited harmonic signal to frequencies above an upper frequency limit of the band-limited harmonics signal, where the non-linear transformation comprises performing a linear convolution; and
inverse transforming, through the use of the processor, the extended complex spectrum of the band-limited harmonic signal back into the time domain.
2. The method of claim 1 where the step of transforming the band-limited harmonic signal from the time domain to the frequency domain comprises performing a Fast Fourier Transform (FFT) on the band-limited harmonic signal.
3. The method of claim 2 where the step of performing the linear convolution comprises performing a linear convolution on the complex spectrum of the band-limited harmonic signal with itself.
4. The method of claim 3 where the linear convolution is performed according to a formula Y(k)=X(k)*X(k); k=0 . . . N/2; where * denotes a linear convolution operation, k is a frequency index and N is a length of the FFT used in transforming the band-limited harmonic signal from the time domain to the frequency domain.
5. The method of claim 2 where the step of performing the linear convolution comprises performing a linear convolution on the spectrum of the band-limited harmonic signal weighted by a first weighting factor and the complex spectrum of the band-limited harmonic signal weighted by a second weighting factor.
6. The method of claim 2 where the step of performing the linear convolution comprises performing a linear convolution of a portion of the complex spectrum of the band-limited harmonic signal with a portion of the complex spectrum of the band-limited harmonic signal.
7. The method of claim 1 where the step of transforming the band-limited harmonic signal from the time domain to the frequency domain comprises employing one of:
a Discrete Fourier Transform (DFT);
a Discrete Cosine Transform (DCT);
a filter bank; or
a wavelet transform.
8. The method of claim 1 , where performing a linear convolution reduces aliasing artifacts.
9. The method of claim 1 , where the linear convolution is performed according to a formula Y(k)=[G 1 (k 1 )X(k 1 )]*[G 2 (k 2 )X(k 2 )]; k 1 =0 . . . M 1 ; k 2 =0 . . . M 2 ; M 1 , M 2 <N/2; where * denotes a linear convolution operation, k 1 is a first frequency index, k 2 is a second frequency index, G 1 is a first weighting factor, G 2 is a second weighting factor, and N is a length of a Fast Fourier Transform used in transforming the band-limited harmonic signal from the time domain to the frequency domain.
10. The method of claim 1 , further comprising where a portion of the band-limited harmonic signal is corrupted by noise; and
where a portion of the complex spectrum of the band-limited harmonic signal corresponding to the portion of the band-limited harmonic signal corrupted by noise is excluded from the complex spectrum of the band-limited harmonic signal before beginning the step of performing, through the use of a processor, a non-linear transformation on the complex spectrum of the band-limited harmonic signal in the frequency domain.
11. A computer-implemented harmonic extension method comprising:
receiving a band-limited harmonic signal having significant signal energies at regular frequency intervals within a limited frequency band defined by a passband lower frequency limit and a passband upper frequency limit;
transforming, through the use of a processor, the band-limited harmonic signal from a time domain to a frequency domain to obtain a complex spectrum of the band-limited harmonic signal;
performing, through the use of the processor, a first non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain which extends harmonic content of the band-limited harmonic signal to frequencies below the passband lower frequency limit where a lower extended spectrum has harmonic energy at at least one harmonic frequency at which harmonic energy was absent in the band-limited harmonic signal,
where the first non-linear transformation comprises performing a linear convolution on the complex spectrum of the band-limited harmonic signal with a mirrored complex conjugate of the complex spectrum of the band-limited harmonic signal;
performing, through the use of the processor, a second non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain which extends harmonic content of the band-limited harmonic signal to frequencies above the passband upper frequency limit where an upper extended spectrum has harmonic energy at at least one harmonic frequency at which harmonic energy was absent in the band-limited harmonic signal,
where the second non-linear transformation comprises performing a linear convolution on the complex spectrum of the band-limited harmonic signal with itself; and
transforming, through the use of the processor, the lower extended spectrum and the upper extended spectrum into the time domain.
12. The method of claim 11 where the step of transforming the band-limited harmonic signal from the time domain to the frequency domain comprises performing a Fast Fourier Transform on the band-limited harmonic signal.
13. The method of claim 12 where the step of performing the linear convolution on the complex spectrum of the band-limited harmonic signal with itself further comprises performing a linear convolution on the spectrum of the band-limited harmonic signal weighted by a first weighting factor and the complex spectrum of the band-limited harmonic signal weighted by a second weighting factor.
14. The method of claim 12 where the step of performing the linear convolution on the complex spectrum of the band-limited harmonic signal with a mirrored complex conjugate of the complex spectrum of the band-limited harmonic signal further comprises performing a linear convolution of a portion of the complex spectrum of the band-limited harmonic signal with a portion of the mirrored complex conjugate of the complex spectrum of the band-limited harmonic signal.
15. The method of claim 11 where the step of transforming the band-limited harmonic signal from the time domain to the frequency domain comprises employing one of:
a Discrete Fourier Transform (DFT);
a Discrete Cosine Transform (DCT);
a digital filter bank; or a wavelet transform.
16. The method of claim 11 , where the linear convolution performed on the complex spectrum of the band-limited harmonic signal with the mirrored complex conjugate of the complex spectrum of the band-limited harmonic signal is performed according to a formula Y(k)=X(k)*conj(X(N-k)) k=0 . . . N/2; where * denotes a linear convolution operation, k is a frequency index and N is a length of a Fast Fourier Transform used in transforming the band-limited harmonic signal from the time domain to the frequency domain.
17. A system for extending harmonics of a band-limited harmonic signal, the system comprising:
means for receiving a band-limited harmonic signal;
a signal processor having a forward transform module that transforms the band-limited harmonic signal from a time domain into a complex spectrum of the band-limited harmonic signal in a frequency domain; and
a harmonic generation module that performs a non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain, the non-linear transformation comprising a linear convolution in the frequency domain that extends harmonic content of the band-limited harmonic signal to a frequency above an upper frequency limit of the band-limited harmonic signal; and
a reverse transform module that transforms the harmonically extended spectrum of the band-limited harmonic signal back to the time domain.
18. The system of claim 17 where the forward transform employed is one of a Fast Fourier Transform (FFT); a Discrete Fourier transform (DFT); a Discrete Cosine Transform (DCT); a digital filter bank; or a wavelet transform to transform the band-limited harmonic signal into the frequency domain.
19. The system of claim 17 where the linear convolution is performed on the complex spectrum of the band-limited harmonic signal with itself.
20. The system of claim 17 where the linear convolution is performed on the complex spectrum of the band-limited harmonic signal weighted by a first weighting factor, with the complex spectrum of the band-limited harmonic signal weighted by a second weighting factor.
21. A system for extending harmonics and a spectral envelope of a band-limited harmonic signal and combining an extended signal with the band-limited harmonic signal, the system comprising:
means for receiving a band-limited harmonic signal;
a signal processor having a forward transform module that transforms the band-limited harmonic signal from a time domain into a complex spectrum of the band-limited harmonic signal in a frequency domain;
a harmonic generation module that performs a non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain that extends harmonic content of the band-limited harmonic signal to frequencies above an upper frequency limit of the band-limited harmonic signal, the non-linear transformation comprising a linear convolution in the frequency domain;
a spectral envelope extender module to ensure the spectral envelope of the harmonically extended spectrum of the band-limited harmonic signal is complimentary to that of the band-limited signal;
a combiner module that combines the harmonically extended spectrum of the band-limited harmonic signal with a spectrum of the band-limited harmonic signal to create a final frequency extended harmonic spectrum; and
a reverse transform module for transforming the final frequency extended harmonic spectrum back to the time domain.
22. The system of claim 21 where the forward transform employed is one of a Fast Fourier Transform (FFT); a Discrete Fourier transform (DFT); a Discrete Cosine Transform (DCT); a digital filter bank; or a wavelet transforms to transform the band-limited harmonic signal into the frequency domain.
23. The system of claim 21 where the linear convolution is performed on the complex spectrum of the band-limited harmonic signal with itself.
24. The system of claim 21 where the linear convolution is performed on the complex spectrum of the band-limited harmonic signal weighted by a first weighting factor with the complex spectrum of the band-limited harmonic signal weighted by a second weighting factor.
25. A system for extending harmonics of a band-limited harmonic signal, the system comprising:
means for receiving a band-limited harmonic signal;
a signal processor having a forward transform module that transforms the band-limited harmonic signal from a time domain into a complex spectrum of the band-limited harmonic signal in a frequency domain; and
a harmonic generation module that performs a first non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain, the first non-linear transformation comprising a first linear convolution in the frequency domain that extends harmonic content of the band-limited harmonic signal to a frequency above an upper frequency limit of the band-limited harmonic signal, and that performs a second non-linear transformation of the complex spectrum of the band-limited harmonic signal in the frequency domain, the second non-linear transformation comprising a second linear convolution in the frequency domain that extends harmonic content of the band-limited harmonic signal to a frequency below a lower frequency limit of the band-limited harmonic signal; and
a reverse transform module that transforms harmonic content of the band-limited harmonic signal extended to the frequency above the upper frequency limit of the band-limited harmonic signal and harmonic content of the band-limited harmonic signal extended to the frequency below the lower frequency limit of the band-limited harmonic signal into the time domain.
26. The system of claim 25 where the first linear convolution comprises a linear convolution of the complex spectrum of the band-limited harmonic signal convolved with itself.
27. The system of claim 25 where the second linear convolution comprises a linear convolution of the complex spectrum of the band-limited harmonic signal convolved with a mirrored complex conjugate of the complex spectrum of the band-limited harmonic signal.