Programmer Guide/Command Reference/EVAL/cepstrum: Difference between revisions

Revision as of 14:20, 12 April 2011

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 the 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)=2^M/2+1).

Result 1
The next (nearest) possible signal window length.

Usage 2
fft(x)
x
signal vector or matrix; if x is a matrix a spectrum of each column is computed

Result 2
A matrix y with 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(nrow(x)).

Usage 3
fft(x, n {, ytype, poffset, prange, aref})
x
signal vector or matrix; if x is a matrix a spectrum of each column is computed
n
desired length of analysis window;
If n < nrow(x), the analysis window length L is set to nrow(x), otherwise L is set to n.
If the analysis window length L is a power of 2 (L=2^M), the fft algorithm is used, otherwise the dft is used.
If L is greater than nrow(x), zero padding is applied to the signal.
ytype
select the type and format of the computed spectrum (default=0) -> see Result 3
poffset
offset in samples to the signal begin or the selected zero phase position (default=0)



poffset=0
-> phase[i] = atan2(im[i], re[i])


otherwise
-> phase[i] = (atan2(im[i], re[i]) - 2*pi*i/L * poffset) % (2 * pi)

prange
selects the range of phase values (default=0)



prange=0
-> 0 <= phase[i] < 2*pi


otherwise
-> -pi <= phase[i] < pi

aref
reference amplitude if the log. spectrum (ytype=4) is requested (default=1)

Result 3
A matrix y with 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.



ytype
description
content of y[*,j]
nrow(y)


0

complex spectrum in cartesian format

{ re₀, im₀, re₁, im₁, ... }

L+2


1

complex spectrum in polar format

{ amp₀, phase₀, amp₁, phase₁, ... }
with: amp_i=sqrt(re_i² + im_i²),
phase_i see poffset

L+2


2

amplitude spectrum

{ amp₀, amp₁, ... }

L+1


3

power spectrum

{ amp₀², amp₁², ... }

L+1


4

logarithmic amplitude spectrum

{ lev₀, lev₁, ... }
with: lev_i=20*log₁₀(amp_i/aref)

L+1

See also
ifft, dft, dct, cepstrum, lpc, complex arithmetic

<function list>

Contents


1 cepstrum
2 Usage:
3 Function:
4 Result:
5 Usage:
6 Function:
7 Result:
8 Usage:
9 Function:
10 Result:
11 Notes:

cepstrum

Cepstrum.

Usage:

cepstrum(x)

Function:

Computes the real cepstrum of x.

Result:

A vector containing the cepstrum coefficients.

Usage:

cepstrum(x, n)

Function:

Computes the cepstrum smoothed spectrum of x using the first n coefficients.

Result:

The cepstrum smoothed spectrum. This is the same type as x.

Usage:

cepstrum(x, o, n)

Function:

Computes the cepstrum smoothed spectrum of x using the coefficients o to o+n-1.

Result:

The cepstrum smoothed spectrum. This is the same type as x.

Notes:

x must be a log. amplitude spectrum computed with a radix-2 fft.

@@ Line 1: / Line 1: @@
+{{DISPLAYTITLE:{{SUBPAGENAME}}}}
+Compute the real cepstrum or the cepstrum smoothed/liftered spectrum of the logarithmic spectrum ''x''.
+----
+;Usage 1:<code>cepstrum(<var>x</var>)</code>
+:;<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>).
+;Result 1:The next (nearest) possible signal window length.
+----
+;Usage 2:<code>fft(<var>x</var>)</code>
+:;<var>x</var>: signal vector or matrix; if ''x'' is a matrix a spectrum of each column is computed
+;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''))]].
+----
+;Usage 3:<code>fft(<var>x</var>, <var>n</var> {, <var>ytype</var>, <var>poffset</var>, <var>prange</var>, <var>aref</var>})</code>
+:;<var>x</var>: signal vector or matrix; if ''x'' is a matrix a spectrum of each column is computed
+:;<var>n</var>: desired length of analysis window;
+::*If <code>''n'' < nrow(''x'')</code>, the analysis window length L is set to nrow(''x''), otherwise L is set to ''n''.
+::*If the analysis window length L is a power of 2 (L=2^M), the '''fft''' algorithm is used, otherwise the '''dft''' is used.
+::*If L is greater than [[../nrow|nrow(''x'')]], zero padding is applied to the signal.
+:;<var>ytype</var>: select the type and format of the computed spectrum (default=0) -> see '''Result 3'''
+:;<var>poffset</var>: offset in samples to the signal begin or the selected ''zero phase'' position (default=0)
+:::{|class="keinrahmen"
+|''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]]
+[[Programmer_Guide/Command_Reference/EVAL#Functions|<function list>]]
 {{DISPLAYTITLE:{{SUBPAGENAME}}}}
 =====cepstrum=====

poffset=0	-> `phase[i] = atan2(im[i], re[i])`
otherwise	-> `phase[i] = (atan2(im[i], re[i]) - 2pii/L * poffset) % (2 * pi)`

prange=0	-> `0 <= phase[i] < 2*pi`
otherwise	-> `-pi <= phase[i] < pi`

Programmer Guide/Command Reference/EVAL/cepstrum: Difference between revisions

Revision as of 14:20, 12 April 2011

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cepstrum

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ytype	description	content of y[*,j]	nrow(y)
0	complex spectrum in cartesian format	`{ re₀, im₀, re₁, im₁, ... }`	L+2
1	complex spectrum in polar format	`{ amp₀, phase₀, amp₁, phase₁, ... } with: amp_i=sqrt(re_i² + im_i²), phase_i see poffset`	L+2
2	amplitude spectrum	`{ amp₀, amp₁, ... }`	L+1
3	power spectrum	`{ amp₀², amp₁², ... }`	L+1
4	logarithmic amplitude spectrum	`{ lev₀, lev₁, ... } with: lev_i=20*log₁₀(amp_i/aref)`	L+1