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

Latest revision as of 19:22, 21 April 2011

Compute the discrete fourier transform of a real signal using the fft or the dft algorithm.

Usage 1

fft(n)

n: desired signal window length (scalar)

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]) - 2pii/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

@@ Line 1: / Line 1: @@
 {{DISPLAYTITLE:{{SUBPAGENAME}}}}
-Fast fourier transform.
+Compute the discrete fourier transform of a real signal using the '''fft''' or the '''dft''' algorithm.
 ----
-;Usage 1:<code>fft(<var>x</var>)</code>
+;Usage 1:<code>fft(<var>n</var>)</code>
+:;<var>n</var>: desired signal window length (scalar)
+;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:A matrix containing a complex spectrum for each column (channel) of the input signal ''x''. The transformation length L is set to <code>[[./npow2|npow2(nrow(
+;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''))]].
-//	(1)	fft( X )
+----
-	//		function:
+;Usage 3:<code>fft(<var>x</var>, <var>n</var> {, <var>ytype</var>, <var>poffset</var>, <var>prange</var>, <var>aref</var>})</code>
-	//			if X is a vector: compute the fft spectrum of X
+:;<var>x</var>: signal vector or matrix; if ''x'' is a matrix a spectrum of each column is computed
-	//			if X is a matrix: compute the fft spectrum of each row of X
+:;<var>n</var>: desired length of analysis window;
-	//			the fft length L is set to the next power of 2 of nrow(X)
+::*If <code>''n'' < nrow(''x'')</code>, the analysis window length L is set to nrow(''x''), otherwise L is set to ''n''.
-	//			if L is greater than nrow(X), zero padding is applied
+::*If the analysis window length L is a power of 2 (L=2<sup>M</sup>), the '''fft''' algorithm is used, otherwise the '''dft''' is used.
-	//		result:
+::*If L is greater than [[../nrow|nrow(''x'')]], zero padding is applied to the signal.
-	//			vector or matrix Y containing one complex spectrum per row;
+:;<var>ytype</var>: select the type and format of the computed spectrum (default=0) &rarr; see '''Result 3'''
-	//			nrow(Y) = L+2, ncol(Y) = ncol(X)
+:;<var>poffset</var>: offset in samples to the signal begin or the selected ''zero phase'' position (default=0)
+:::{|class="keinrahmen"
+|''poffset''='''0''' ||&rarr; <code>phase[i] = atan2(im[i], re[i])</code>
+|-
-=====fft=====
+|otherwise ||&rarr; <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''' ||&rarr; <code>0 &le; phase[i] < 2*pi</code>
+|-
+|otherwise ||&rarr; <code>-pi &le; 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
 |-
-|fft(x)fft(<var>x</var>, <var>n</var>)
+|'''4'''
-|Calculate the FFT of the vector x. If the parameter <var>n</var> is given and <var>n</var> >= nrow(<var>x</var>), then <var>n</var> is used as the signal length. The next possible value of <var>x</var>^2 is used for the FFT length (if necessary, 0 samples are added to <var>x</var>). The result is a complex FFT spectrum with 2^(m-1)+1 complex values (2^m = FFT length)
+|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: [[../ifft|ifft]], [[../dft|dft]], [[../dct|dct]], [[../cepstrum|cepstrum]], [[../lpc|lpc]], [[../complex arithmetic|complex arithmetic]]
+[[../#Functions|<function list>]]

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

Latest revision as of 19:22, 21 April 2011

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