The new SignalProcessing:-Quantize command is used to replace real-valued data in a container with values from a codebook.

For example, suppose we want to quantize the values generated by $\sin \left(t\right)$ over the interval $\left[0\,2\mathrm{\pi }\right]$ so that the values are to the nearest quarter integer. With this new command, we can do either of the following:

X := GenerateSignal( sin(t), t = 0 .. 2 * Pi, 50 )^+;

$X≔\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc}0.& 0.127877161684506& 0.253654583909507& 0.375267004879374& 0.490717552003938& 0.598110530491216& 0.695682550603486& 0.781831482468030& 0.855142763005346& 0.914412623015812& 0.958667853036661& 0.987181783414450& 0.999486216200688& 0.995379112949198& 0.974927912181824& 0.938468422049760& 0.886599306373000& 0.820172254596956& 0.740277997075315& 0.648228395307788& 0.545534901210549& 0.433883739117558& 0.315108218023621& 0.191158628701372& 0.0640702199807128& \mathrm{-0.0640702199807130}& \mathrm{-0.191158628701372}& \mathrm{-0.315108218023621}& \mathrm{-0.433883739117558}& \mathrm{-0.545534901210548}& \mathrm{-0.648228395307789}& \mathrm{-0.740277997075316}& \mathrm{-0.820172254596956}& \mathrm{-0.886599306373000}& \mathrm{-0.938468422049761}& \mathrm{-0.974927912181824}& \mathrm{-0.995379112949198}& \mathrm{-0.999486216200688}& \mathrm{-0.987181783414450}& \mathrm{-0.958667853036660}& \mathrm{-0.914412623015812}& \mathrm{-0.855142763005346}& \mathrm{-0.781831482468030}& \mathrm{-0.695682550603486}& \mathrm{-0.598110530491216}& \mathrm{-0.490717552003938}& \mathrm{-0.375267004879374}& \mathrm{-0.253654583909508}& \mathrm{-0.127877161684506}& \mathrm{-2.44929359829471}\times {10}^{\mathrm{-16}}\end{array}\right]$

Y := Quantize( X, -1 .. 1, 9 );

$Y≔\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc}0.& 0.250000000000000& 0.250000000000000& 0.500000000000000& 0.500000000000000& 0.500000000000000& 0.750000000000000& 0.750000000000000& 0.750000000000000& 1.& 1.& 1.& 1.& 1.& 1.& 1.& 1.& 0.750000000000000& 0.750000000000000& 0.750000000000000& 0.500000000000000& 0.500000000000000& 0.250000000000000& 0.250000000000000& 0.& 0.& \mathrm{-0.250000000000000}& \mathrm{-0.250000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.250000000000000}& \mathrm{-0.250000000000000}& 0.\end{array}\right]$

Z := Quantize( X, < seq( 0.25 * k, k = -4 .. 4 ) > );

$Z≔\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc}0.& 0.250000000000000& 0.250000000000000& 0.500000000000000& 0.500000000000000& 0.500000000000000& 0.750000000000000& 0.750000000000000& 0.750000000000000& 1.& 1.& 1.& 1.& 1.& 1.& 1.& 1.& 0.750000000000000& 0.750000000000000& 0.750000000000000& 0.500000000000000& 0.500000000000000& 0.250000000000000& 0.250000000000000& 0.& 0.& \mathrm{-0.250000000000000}& \mathrm{-0.250000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-1.}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.750000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.500000000000000}& \mathrm{-0.250000000000000}& \mathrm{-0.250000000000000}& 0.\end{array}\right]$

The command can also display the original and quantized signals together:

Quantize( X, -1 .. 1, 9, 'output' = 'plot', 'emphasizecodes' );

The example above uses the default nearest for the option truncation, but below and above are also available. Moreover, classical quantization techniques midriser and midtread are provided:

Quantize( X, 'midriser', 1, 9, 'output' = 'plot', 'emphasizecodes' );

The SignalProcessing:-GenerateSignal command has also been updated to include the quantization option:

GenerateSignal( sin(t), t = 0 .. 2 * Pi, 50, 'quantization' = [-1..1,9], 'output' = 'signalplot' );

The new SignalProcessing:-SavitzkyGolayFilter command applies the Savitzky-Golay Filter to a signal, with applications including smoothing noisy data and estimating derivatives of data. The filter is also known as Polynomial Smoothing, Least-Squares Smoothing, and Locally Weighted Scatterplot Smoothing (LOWESS).

For example, consider the following:

f := sin(t) + 1/5 * cos(3*t); # original expression

$f≔\sin \left(t\right)+\frac{\cos \left(3t\right)}{5}$

t1 := 0; # start time

$\mathrm{t1}≔0$

t2 := 2 * Pi; # finish time

$\mathrm{t2}≔2\mathrm{\pi }$

m := 200; # number of points

$m≔200$

(T,X) := GenerateSignal( f, t = t1 .. t2, m, 'noisedeviation' = 0.2, 'includefinishtime' = 'false', 'output' = ['times','signal'] );

The smoothing, both unweighted and weighted, noticeably reduces the noise:

d := 2; # degree of polynomial interpolants

$d≔2$

r := 10; # radius of frame

$r≔10$

W := < seq( 1 .. r + 1 ), seq( r .. 1, -1 ) >: # weights

SavitzkyGolayFilter( X, d, r, 'timerange' = t1 .. t2, 'plotoptions' = [ 'title' = "Unweighted Savitzky-Golay Filtered Signal" ], 'output' = 'plot' );

SavitzkyGolayFilter( X, d, W, 'timerange' = t1 .. t2, 'plotoptions' = [ 'title' = "Weighted Savitzky-Golay Filtered Signal" ], 'output' = 'plot' );

The SignalProcessing:-DifferentiateData command now includes an option to estimate the derivative of a signal using the Savitzky-Golay Filter;

(dt,T,U) := GenerateSignal( t * (t^2-1)^3, t = -1 .. 1, 75, 'includefinishtime' = 'false', 'output' = ['timestep','times','signal'] );

V := DifferentiateData( U, 1, 'step' = dt, 'method' = 'savitzkygolay', 'extrapolation' = 'periodic', 'frameradius' = 2 );

dataplot( T, [U,V], 'style' = 'line', 'legend' = ["Signal","First derivative"], 'color' = ['red','blue'], 'view' = [-1..1,'DEFAULT'] );

The SignalProcessing:-Convolution command has been updated to include shape options full (the default), same, and valid. For example:

A := LinearAlgebra:-RandomVector( 5, 'generator' = -1.0 .. 1.0, 'datatype' = 'float[8]' );

$A≔\left[\begin{array}{c}\mathrm{-0.844885936681858}\\ 0.604222951371852\\ \mathrm{-0.434653429149560}\\ \mathrm{-0.809289661861375}\\ 0.964721117374344\end{array}\right]$

B := LinearAlgebra:-RandomVector( 3, 'generator' = -1.0 .. 1.0, 'datatype' = 'float[8]' );

$B≔\left[\begin{array}{c}0.360574124053584\\ \mathrm{-0.983812180981714}\\ 0.254768677798811\end{array}\right]$

C1 := Convolution( A, B, 'shape' = 'full' );

$\mathrm{C1}≔\left[\begin{array}{ccccccc}\mathrm{-0.304644006544253}& 1.04907623747173& \mathrm{-0.966417152050085}& 0.289745509588224& 1.03330641968989& \mathrm{-1.15528604363184}& 0.245780723518053\end{array}\right]$

C2 := Convolution( A, B, 'shape' = 'same' );

$\mathrm{C2}≔\left[\begin{array}{ccccc}1.04907623747173& \mathrm{-0.966417152050085}& 0.289745509588224& 1.03330641968989& \mathrm{-1.15528604363184}\end{array}\right]$

C3 := Convolution( A, B, 'shape' = 'valid' );

$\mathrm{C3}≔\left[\begin{array}{ccc}\mathrm{-0.966417152050085}& 0.289745509588224& 1.03330641968989\end{array}\right]$

Moreover, lists can now be passed for signals in addition to rtables.

The SignalProcessing:-FFT and SignalProcessing:-InverseFFT commands now support padding and truncation. Padding is helpful due to the increase in both the speed of computation and the frequency resolution. The size option accepts a positive integer for the new size and keywords same (the default, for not changing the length) and recommended (for changing the length to the next power of 2). For example:

X := LinearAlgebra:-RandomVector( 3, 'generator' = -1.0 .. 1.0, 'datatype' = 'complex[8]' );

$X≔\left[\begin{array}{c}\mathrm{-0.235334127009411}+0.524842563866319\mathrm{I}\\ \mathrm{-0.600897572737194}-0.723997348520040\mathrm{I}\\ 0.0678662154384722-0.122666338816995\mathrm{I}\end{array}\right]$

Y1 := FFT( X, 'size' = 'same' );

$\mathrm{Y1}≔\left[\begin{array}{c}\mathrm{-0.443616019201318}-0.185803512266725\mathrm{I}\\ \mathrm{-0.282662827618448}+0.881810643516785\mathrm{I}\\ 0.318668182084598+0.213046855341119\mathrm{I}\end{array}\right]$

Y2 := FFT( X, 'size' = 'recommended' );

$\mathrm{Y2}≔\left[\begin{array}{c}\mathrm{-0.384182742154067}-0.160910561735358\mathrm{I}\\ \mathrm{-0.513598845483962}+0.624203237710254\mathrm{I}\\ 0.216714830583127+0.563086786784682\mathrm{I}\\ 0.210398503036078+0.0233056649730601\mathrm{I}\end{array}\right]$

Y3 := FFT( X, 'size' = 5 );

$\mathrm{Y3}≔\left[\begin{array}{c}\mathrm{-0.343623490895507}-0.143922781735175\mathrm{I}\\ \mathrm{-0.553020070919267}+0.416781168114568\mathrm{I}\\ \mathrm{-0.0165998120551015}+0.666529860535818\mathrm{I}\\ 0.259681898881381+0.292888860328475\mathrm{I}\\ 0.127338369569880-0.0586934569533205\mathrm{I}\end{array}\right]$

The size option also works with multi-dimensional input. Moreover, lists can now be passed for signals in addition to rtables.