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| ;Usage 1:<code>cepstrum(<var>x</var>)</code> | | ;Usage 1:<code>cepstrum(<var>x</var>)</code> |
| :;<var>x</var>: a vector containing a log. amplitude spectrum | | :;<var>x</var>: a vector containing a log. amplitude spectrum |
| :::*note: because the fft/ifft is used for the computation, the spectrum stored in ''x'' must be computed from a signal with <code>L=2^M</samples> (<code>-> nrow(''x'')=2^M/2+1</code>).
| | ::*note: because a radix-2 fft/ifft is used for the computation, the spectrum stored in ''x'' must be computed from a signal with <code>L=2^M</code> samples (<code>→ nrow(''x'')=L/2+1=2^M/2+1</code>). |
| ;Result 1:The next (nearest) possible signal window length. | | ;Result 1:A vector with <code>L/2</code> values, containing the coefficients or the real cepstrum of ''x''. |
| ---- | | ---- |
| ;Usage 2:<code>fft(<var>x</var>)</code> | | ;Usage 2:<code>cepstrum(<var>x</var>, <var>m</var>)</code> |
| :;<var>x</var>: signal vector or matrix; if ''x'' is a matrix a spectrum of each column is computed | | :;<var>x</var>: a vector containing a log. amplitude spectrum |
| ;Result 2:A matrix ''y'' with [[../ncol|ncol(''x'')]] columns and L+2 rows, where each column ''y''[*,j] contains the complex spectrum of the column (channel) ''x''[*,j]. The transformation length L is set to [[../npow2|npow2(nrow(''x''))]]. | | :;<var>m</var>: the number of coefficients to be used for smoothing; 0 < ''m'' ≤ L/2 |
| | ;Description: |
| | :# The real cepstrum ''c'' of ''x'' is computed. |
| | :# The coefficents ''m'' to <code>L/2-1</code> of the cepstrum ''c'' are set to zero. |
| | :# The smoothed spectrum is computed from the modified cepstrum ''c''. |
| | ;Result 2:A vector with same length as ''x'', containing the smoothed spectrum. |
| ---- | | ---- |
| ;Usage 3:<code>fft(<var>x</var>, <var>n</var> {, <var>ytype</var>, <var>poffset</var>, <var>prange</var>, <var>aref</var>})</code> | | ;Usage 2:<code>cepstrum(<var>x</var>, <var>o</var>, <var>m</var>)</code> |
| :;<var>x</var>: signal vector or matrix; if ''x'' is a matrix a spectrum of each column is computed | | :;<var>x</var>: a vector containing a log. amplitude spectrum |
| :;<var>n</var>: desired length of analysis window; | | :;<var>o</var>: the first coefficient to be used for smoothing; 0 < ''o'' < L/2 |
| ::*If <code>''n'' < nrow(''x'')</code>, the analysis window length L is set to nrow(''x''), otherwise L is set to ''n''.
| | :;<var>m</var>: the number of coefficients to be used for smoothing; 0 < ''m'' ≤ L/2-''o'' |
| ::*If the analysis window length L is a power of 2 (L=2^M), the '''fft''' algorithm is used, otherwise the '''dft''' is used.
| | ;Description: |
| ::*If L is greater than [[../nrow|nrow(''x'')]], zero padding is applied to the signal.
| | :# The real cepstrum ''c'' of ''x'' is computed. |
| :;<var>ytype</var>: select the type and format of the computed spectrum (default=0) -> see '''Result 3''' | | :# The coefficents 0 to ''o''-1 and ''o''+''m'' to <code>L/2-1</code> of the cepstrum ''c'' are set to zero. |
| :;<var>poffset</var>: offset in samples to the signal begin or the selected ''zero phase'' position (default=0)
| | :# The liftered spectrum is computed from the modified cepstrum ''c''. |
| :::{|class="keinrahmen"
| | ;Result 2:A vector with same length as ''x'', containing the liftered spectrum. |
| |''poffset''='''0''' ||-> <code>phase[i] = atan2(im[i], re[i])</code>
| |
| |-
| |
| |otherwise ||-> <code>phase[i] = (atan2(im[i], re[i]) - 2*pi*i/L * ''poffset'') % (2 * pi)</code>
| |
| |}
| |
| :;<var>prange</var>: selects the range of phase values (default=0)
| |
| :::{|class="keinrahmen"
| |
| |''prange''='''0''' ||-> <code>0 <= phase[i] < 2*pi</code>
| |
| |-
| |
| |otherwise ||-> <code>-pi <= phase[i] < pi</code>
| |
| |}
| |
| :;<var>aref</var>: reference amplitude if the log. spectrum (''ytype''=4) is requested (default=1)
| |
| ;Result 3: A matrix ''y'' with [[../ncol|ncol(''x'')]] columns, where each column ''y''[*,j] contains the spectrum of the column (channel) ''x''[*,j]. The type and the length of the spectra is selected by the argument ''ytype''.
| |
| ::{|class="einrahmen" | |
| !''ytype'' !! description !! content of ''y''[*,j] !! nrow(''y'')
| |
| |-
| |
| |'''0'''
| |
| |complex spectrum in cartesian format
| |
| |<code>{ re<sub>0</sub>, im<sub>0</sub>, re<sub>1</sub>, im<sub>1</sub>, ... }</code>
| |
| |L+2
| |
| |-
| |
| |'''1'''
| |
| |complex spectrum in polar format
| |
| |<code>{ amp<sub>0</sub>, phase<sub>0</sub>, amp<sub>1</sub>, phase<sub>1</sub>, ... }<BR>with: amp<sub>i</sub>=sqrt(re<sub>i</sub><sup>2</sup> + im<sub>i</sub><sup>2</sup>),<BR>phase<sub>i</sub> see ''poffset''</code>
| |
| |L+2
| |
| |-
| |
| |'''2'''
| |
| |amplitude spectrum
| |
| |<code>{ amp<sub>0</sub>, amp<sub>1</sub>, ... }</code>
| |
| |L+1
| |
| |-
| |
| |'''3'''
| |
| |power spectrum
| |
| |<code>{ amp<sub>0</sub><sup>2</sup>, amp<sub>1</sub><sup>2</sup>, ... }</code>
| |
| | L+1
| |
| |-
| |
| |'''4'''
| |
| |logarithmic amplitude spectrum
| |
| |<code>{ lev<sub>0</sub>, lev<sub>1</sub>, ... }<BR>with: lev<sub>i</sub>=20*log<sub>10</sub>(amp<sub>i</sub>/''aref'')</code>
| |
| | L+1
| |
| |}
| |
| ---- | | ---- |
| ;See also: [[Programmer_Guide/Command_Reference/EVAL/ifft|ifft]], [[Programmer_Guide/Command_Reference/EVAL/dft|dft]], [[Programmer_Guide/Command_Reference/EVAL/dct|dct]], [[Programmer_Guide/Command_Reference/EVAL/cepstrum|cepstrum]], [[Programmer_Guide/Command_Reference/EVAL/lpc|lpc]], [[Programmer_Guide/Command_Reference/EVAL/complex arithmetic|complex arithmetic]] | | ;See also: [[../fft|fft]], [[../ifft|ifft]], [[../dft|dft]], [[../dct|dct]], [[../lpc|lpc]], [[../ipeak|ipeak]], [[../formants|formants]] |
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| [[Programmer_Guide/Command_Reference/EVAL#Functions|<function list>]] | | [[../#Functions|<function list>]] |
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| {{DISPLAYTITLE:{{SUBPAGENAME}}}}
| |
| =====cepstrum=====
| |
| | |
| Cepstrum.
| |
| | |
| =====Usage:=====
| |
| | |
| <code>cepstrum(<var>x</var>)</code>
| |
| | |
| =====Function:=====
| |
| | |
| Computes the real cepstrum of <var>x</var>.
| |
| | |
| =====Result:=====
| |
| | |
| A vector containing the cepstrum coefficients.
| |
| | |
| =====Usage:=====
| |
| | |
| <code>cepstrum(<var>x</var>, <var>n</var>)</code>
| |
| | |
| =====Function:=====
| |
| | |
| Computes the cepstrum smoothed spectrum of <var>x</var> using the first <var>n</var> coefficients.
| |
| | |
| =====Result:=====
| |
| | |
| The cepstrum smoothed spectrum. This is the same type as <var>x</var>.
| |
| | |
| =====Usage:=====
| |
| | |
| <code>cepstrum(<var>x</var>, <var>o</var>, <var>n</var>)</code>
| |
| | |
| =====Function:=====
| |
| | |
| Computes the cepstrum smoothed spectrum of <var>x</var> using the coefficients o to <var>o</var><code>+</code><var>n</var><code>-1</code>.
| |
| | |
| =====Result:=====
| |
| | |
| The cepstrum smoothed spectrum. This is the same type as <var>x</var>.
| |
| | |
| =====Notes:=====
| |
| | |
| <var>x</var> must be a log. amplitude spectrum computed with a radix-2 fft.
| |
Compute the real cepstrum or the cepstrum smoothed/liftered spectrum of the logarithmic spectrum x.
- Usage 1
cepstrum(x)
- x
- a vector containing a log. amplitude spectrum
- note: because a radix-2 fft/ifft is used for the computation, the spectrum stored in x must be computed from a signal with
L=2^M samples (→ nrow(x)=L/2+1=2^M/2+1).
- Result 1
- A vector with
L/2 values, containing the coefficients or the real cepstrum of x.
- Usage 2
cepstrum(x, m)
- x
- a vector containing a log. amplitude spectrum
- m
- the number of coefficients to be used for smoothing; 0 < m ≤ L/2
- Description
-
- The real cepstrum c of x is computed.
- The coefficents m to
L/2-1 of the cepstrum c are set to zero.
- The smoothed spectrum is computed from the modified cepstrum c.
- Result 2
- A vector with same length as x, containing the smoothed spectrum.
- Usage 2
cepstrum(x, o, m)
- x
- a vector containing a log. amplitude spectrum
- o
- the first coefficient to be used for smoothing; 0 < o < L/2
- m
- the number of coefficients to be used for smoothing; 0 < m ≤ L/2-o
- Description
-
- The real cepstrum c of x is computed.
- The coefficents 0 to o-1 and o+m to
L/2-1 of the cepstrum c are set to zero.
- The liftered spectrum is computed from the modified cepstrum c.
- Result 2
- A vector with same length as x, containing the liftered spectrum.
- See also
- fft, ifft, dft, dct, lpc, ipeak, formants
<function list>
