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1 @c Copyright (C) 1996, 1997 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Signal Processing, Image Processing, Control Theory, Top |
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6 @chapter Signal Processing |
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7 |
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8 I hope that someday Octave will include more signal processing |
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9 functions. If you would like to help improve Octave in this area, |
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10 please contact @email{bug-octave@@bevo.che.wisc.edu}. |
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11 |
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12 @deftypefn {Function File} {} detrend (@var{x}, @var{p}) |
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13 If @var{x} is a vector, @code{detrend (@var{x}, @var{p})} removes the |
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14 best fit of a polynomial of order @var{p} from the data @var{x}. |
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15 |
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16 If @var{x} is a matrix, @code{detrend (@var{x}, @var{p})} does the same |
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17 for each column in @var{x}. |
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18 |
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19 The second argument is optional. If it is not specified, a value of 1 |
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20 is assumed. This corresponds to removing a linear trend. |
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21 @end deftypefn |
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22 |
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23 @deftypefn {Function} {} fft (@var{a}, @var{n}) |
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24 Compute the FFT of @var{a} using subroutines from @sc{Fftpack}. If @var{a} |
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25 is a matrix, @code{fft} computes the FFT for each column of @var{a}. |
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26 |
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27 If called with two arguments, @var{n} is expected to be an integer |
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28 specifying the number of elements of @var{a} to use. If @var{a} is a |
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29 matrix, @var{n} specifies the number of rows of @var{a} to use. If |
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30 @var{n} is larger than the size of @var{a}, @var{a} is resized and |
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31 padded with zeros. |
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32 @end deftypefn |
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33 |
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34 @deftypefn {Loadable Function} {} ifft (@var{a}, @var{n}) |
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35 Compute the inverse FFT of @var{a} using subroutines from @sc{Fftpack}. If |
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36 @var{a} is a matrix, @code{fft} computes the inverse FFT for each column |
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37 of @var{a}. |
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38 |
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39 If called with two arguments, @var{n} is expected to be an integer |
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40 specifying the number of elements of @var{a} to use. If @var{a} is a |
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41 matrix, @var{n} specifies the number of rows of @var{a} to use. If |
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42 @var{n} is larger than the size of @var{a}, @var{a} is resized and |
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43 padded with zeros. |
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44 @end deftypefn |
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45 |
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46 @deftypefn {Loadable Function} {} fft2 (@var{a}, @var{n}, @var{m}) |
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47 Compute the two dimensional FFT of @var{a}. |
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48 |
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49 The optional arguments @var{n} and @var{m} may be used specify the |
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50 number of rows and columns of @var{a} to use. If either of these is |
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51 larger than the size of @var{a}, @var{a} is resized and padded with |
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52 zeros. |
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53 @end deftypefn |
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54 |
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55 @deftypefn {Loadable Function} {} ifft2 (@var{a}, @var{n}, @var{m}) |
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56 Compute the two dimensional inverse FFT of @var{a}. |
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57 |
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58 The optional arguments @var{n} and @var{m} may be used specify the |
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59 number of rows and columns of @var{a} to use. If either of these is |
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60 larger than the size of @var{a}, @var{a} is resized and padded with |
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61 zeros. |
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62 @end deftypefn |
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63 |
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64 @deftypefn {Built-in Function} {} fftconv (@var{a}, @var{b}, @var{n}) |
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65 Return the convolution of the vectors @var{a} and @var{b}, as a vector |
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66 with length equal to the @code{length (a) + length (b) - 1}. If @var{a} |
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67 and @var{b} are the coefficient vectors of two polynomials, the returned |
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68 value is the coefficient vector of the product polynomial. |
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69 |
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70 The computation uses the FFT by calling the function @code{fftfilt}. If |
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71 the optional argument @var{n} is specified, an N-point FFT is used. |
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72 @end deftypefn |
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73 |
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74 @deftypefn {Function File} {} fftfilt (@var{b}, @var{x}, @var{n}) |
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75 |
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76 With two arguments, @code{fftfilt} filters @var{x} with the FIR filter |
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77 @var{b} using the FFT. |
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78 |
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79 Given the optional third argument, @var{n}, @code{fftfilt} uses the |
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80 overlap-add method to filter @var{x} with @var{b} using an N-point FFT. |
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81 @end deftypefn |
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82 |
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83 @deftypefn {Loadable Function} {y =} filter (@var{b}, @var{a}, @var{x}) |
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84 Return the solution to the following linear, time-invariant difference |
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85 equation: |
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86 @iftex |
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87 @tex |
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88 $$ |
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89 \sum_{k=0}^N a_{k+1} y_{n-k} = \sum_{k=0}^M b_{k+1} x_{n-k}, \qquad |
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90 1 \le n \le P |
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91 $$ |
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92 @end tex |
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93 @end iftex |
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94 @ifinfo |
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95 |
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96 @smallexample |
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97 N M |
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98 SUM a(k+1) y(n-k) = SUM b(k+1) x(n-k) for 1<=n<=length(x) |
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99 k=0 k=0 |
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100 @end smallexample |
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101 @end ifinfo |
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102 |
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103 @noindent |
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104 where |
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105 @ifinfo |
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106 N=length(a)-1 and M=length(b)-1. |
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107 @end ifinfo |
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108 @iftex |
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109 @tex |
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110 $a \in \Re^{N-1}$, $b \in \Re^{M-1}$, and $x \in \Re^P$. |
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111 @end tex |
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112 @end iftex |
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113 An equivalent form of this equation is: |
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114 @iftex |
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115 @tex |
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116 $$ |
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117 y_n = -\sum_{k=1}^N c_{k+1} y_{n-k} + \sum_{k=0}^M d_{k+1} x_{n-k}, \qquad |
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118 1 \le n \le P |
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119 $$ |
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120 @end tex |
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121 @end iftex |
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122 @ifinfo |
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123 |
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124 @smallexample |
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125 N M |
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126 y(n) = - SUM c(k+1) y(n-k) + SUM d(k+1) x(n-k) for 1<=n<=length(x) |
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127 k=1 k=0 |
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128 @end smallexample |
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129 @end ifinfo |
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130 |
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131 @noindent |
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132 where |
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133 @ifinfo |
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134 c = a/a(1) and d = b/a(1). |
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135 @end ifinfo |
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136 @iftex |
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137 @tex |
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138 $c = a/a_1$ and $d = b/a_1$. |
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139 @end tex |
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140 @end iftex |
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141 |
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142 In terms of the z-transform, y is the result of passing the discrete- |
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143 time signal x through a system characterized by the following rational |
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144 system function: |
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145 @iftex |
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146 @tex |
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147 $$ |
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148 H(z) = {\displaystyle\sum_{k=0}^M d_{k+1} z^{-k} |
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149 \over 1 + \displaystyle\sum_{k+1}^N c_{k+1} z^{-k}} |
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150 $$ |
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151 @end tex |
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152 @end iftex |
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153 @ifinfo |
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154 |
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155 @example |
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156 M |
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157 SUM d(k+1) z^(-k) |
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158 k=0 |
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159 H(z) = ---------------------- |
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160 N |
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161 1 + SUM c(k+1) z(-k) |
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162 k=1 |
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163 @end example |
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164 @end ifinfo |
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165 @end deftypefn |
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166 |
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167 @deftypefn {Loadable Function} {[@var{y}, @var{sf}] =} filter (@var{b}, @var{a}, @var{x}, @var{si}) |
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168 This is the same as the @code{filter} function described above, except |
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169 that @var{si} is taken as the initial state of the system and the final |
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170 state is returned as @var{sf}. The state vector is a column vector |
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171 whose length is equal to the length of the longest coefficient vector |
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172 minus one. If @var{si} is not set, the initial state vector is set to |
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173 all zeros. |
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174 @end deftypefn |
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175 |
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176 @deftypefn {Function File} {[@var{h}, @var{w}] =} freqz (@var{b}, @var{a}, @var{n}, "whole") |
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177 Return the complex frequency response @var{h} of the rational IIR filter |
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178 whose numerator and denominator coefficients are @var{b} and @var{a}, |
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179 respectively. The response is evaluated at @var{n} angular frequencies |
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180 between 0 and |
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181 @ifinfo |
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182 2*pi. |
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183 @end ifinfo |
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184 @iftex |
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185 @tex |
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186 $2\pi$. |
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187 @end tex |
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188 @end iftex |
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189 |
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190 @noindent |
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191 The output value @var{w} is a vector of the frequencies. |
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192 |
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193 If the fourth argument is omitted, the response is evaluated at |
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194 frequencies between 0 and |
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195 @ifinfo |
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196 pi. |
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197 @end ifinfo |
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198 @iftex |
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199 @tex |
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200 $\pi$. |
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201 @end tex |
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202 @end iftex |
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203 |
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204 If @var{n} is omitted, a value of 512 is assumed. |
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205 |
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206 If @var{a} is omitted, the denominator is assumed to be 1 (this |
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207 corresponds to a simple FIR filter). |
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208 |
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209 For fastest computation, @var{n} should factor into a small number of |
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210 small primes. |
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211 @end deftypefn |
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212 |
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213 @deftypefn {Function File} {} sinc (@var{x}) |
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214 Return |
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215 @iftex |
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216 @tex |
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217 $ \sin (\pi x)/(\pi x)$. |
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218 @end tex |
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219 @end iftex |
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220 @ifinfo |
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221 sin(pi*x)/(pi*x). |
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222 @end ifinfo |
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223 @end deftypefn |
