--- title: "DifferentiatorFilter" language: "en" type: "Symbol" summary: "DifferentiatorFilter[data, \\[Omega]c] applies a differentiator filter with a cutoff frequency \\[Omega]c to an array of data. DifferentiatorFilter[data, \\[Omega]c, n] uses a filter kernel of length n. DifferentiatorFilter[data, \\[Omega]c, n, wfun] applies a smoothing window wfun to the filter kernel." keywords: - differentiator filter - image filter - data filter - sound filter - sound derivative - derivative canonical_url: "https://reference.wolfram.com/language/ref/DifferentiatorFilter.html" source: "Wolfram Language Documentation" related_guides: - title: "Linear and Nonlinear Filters" link: "https://reference.wolfram.com/language/guide/LinearAndNonlinearFilters.en.md" - title: "Image Filtering & Neighborhood Processing" link: "https://reference.wolfram.com/language/guide/ImageFilteringAndNeighborhoodProcessing.en.md" - title: "Signal Filtering & Filter Design" link: "https://reference.wolfram.com/language/guide/SignalFilteringAndFilterDesign.en.md" - title: "Video Computation: Update History" link: "https://reference.wolfram.com/language/guide/VideoComputation-UpdateHistory.en.md" related_functions: - title: "LowpassFilter" link: "https://reference.wolfram.com/language/ref/LowpassFilter.en.md" - title: "HilbertFilter" link: "https://reference.wolfram.com/language/ref/HilbertFilter.en.md" - title: "LeastSquaresFilterKernel" link: "https://reference.wolfram.com/language/ref/LeastSquaresFilterKernel.en.md" - title: "ListConvolve" link: "https://reference.wolfram.com/language/ref/ListConvolve.en.md" - title: "ImageConvolve" link: "https://reference.wolfram.com/language/ref/ImageConvolve.en.md" - title: "DerivativeFilter" link: "https://reference.wolfram.com/language/ref/DerivativeFilter.en.md" - title: "GaussianFilter" link: "https://reference.wolfram.com/language/ref/GaussianFilter.en.md" --- # DifferentiatorFilter DifferentiatorFilter[data, ωc] applies a differentiator filter with a cutoff frequency ωc to an array of data. DifferentiatorFilter[data, ωc, n] uses a filter kernel of length n. DifferentiatorFilter[data, ωc, n, wfun] applies a smoothing window wfun to the filter kernel. ## Details and Options * ``DifferentiatorFilter`` is a finite impulse response (FIR) discrete-time filter that is commonly used to approximate the derivative of sampled data. [image] * The ``data`` can be any of the following: | | | | ------- | ------------------------------------------------- | | list | arbitrary-rank numerical array | | tseries | temporal data such as TimeSeries and TemporalData | | image | arbitrary Image or Image3D object | | audio | an Audio or Sound object | | video | a Video object | * When applied to images and multidimensional arrays, filtering is applied successively to each dimension, starting at level 1. ``DifferentiatorFilter[data, {ωc1, ωc2, …}]`` uses the frequency ``ωci`` for the ``i``$$^{\text{th}}$$ dimension. * Filtering with cutoff frequency ``ωc`` reduces the susceptibility of the derivative to signal noise, with the amount of smoothing dependent on the value of the cutoff frequency ``ωc``. * The cutoff frequency ``ωc`` should be between 0 and $\pi$. Smaller values of ``ωc`` result in greater smoothing. * ``DifferentiatorFilter[data, ωc]`` uses a filter kernel length and smoothing window suitable for the cutoff frequency ``ωc`` and the input ``data``. * Typical smoothing windows ``wfun`` include: | | | | --- | --- | | BlackmanWindow | smoothing with a Blackman window | | DirichletWindow | no smoothing | | HammingWindow | smoothing with a Hamming window | | {v1, v2, …} | use a window with values vi | | f | create a window by sampling f between $-1/2$ and $1/2$ | * The following options can be given: | | | | | ----------- | --------- | --------------------------------- | | Padding | "Fixed" | the padding value to use | | SampleRate | Automatic | sample rate assumed for the input | * By default, ``SampleRate -> 1`` is assumed for images as well as data. For sampled ``sound`` objects with sample rate ``r``, ``SampleRate -> r`` is used. * With ``SampleRate -> r``, the cutoff frequency ``ωc`` should be between 0 and ``r``×$\pi$. --- ## Examples (26) ### Basic Examples (2) Differentiate a sampled cosine signal: ```wl In[1]:= dx = 0.1; data = Table[Cos[2x], {x, 0, 2π, dx}]; ListLinePlot[{data, (1/ 2dx) DifferentiatorFilter[data, Pi, 20]}] Out[1]= [image] ``` --- Differentiate an image: ```wl In[1]:= DifferentiatorFilter[[image], Pi / 2]//ImageAdjust Out[1]= [image] ``` ### Scope (10) #### Data (7) Filter a 1D triangular sequence: ```wl In[1]:= data = BoxMatrix[5, {41}]; ListLinePlot[{data, DifferentiatorFilter[data, 1.]}, PlotRange -> All] Out[1]= [image] ``` --- Filter a 2D pulse sequence: ```wl In[1]:= data = Table[UnitBox[(n/7), (m/7)], {n, -7, 7}, {m, -7, 7}]; ListPlot3D[data] Out[1]= [image] In[2]:= ListPlot3D[DifferentiatorFilter[data, 1.], PlotRange -> All] Out[2]= [image] ``` --- Filter a ``TimeSeries`` : ```wl In[1]:= ts = TemporalData[TimeSeries, {CompressedData["«1188»"], {{0, 1., 0.01}}, 1, {"Continuous", 1}, {"Continuous", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, False, 10.1]; filtered = DifferentiatorFilter[ts, Quantity[10, "Hertz"]]; In[2]:= ListLinePlot[{ts, 5filtered}, PlotLegends -> {"original", "filtered"}, PlotRange -> All] Out[2]= [image] ``` --- Digital differentiation of a square wave audio signal: ```wl In[1]:= a = AudioGenerator["Square"] Out[1]= \!\(\*AudioBox["![Embedded Audio Player](audio://content-gsfre)"]\) In[2]:= DifferentiatorFilter[a, Quantity[15000, "Hertz"]] Out[2]= \!\(\*AudioBox["![Embedded Audio Player](audio://content-jxkl1)"]\) ``` --- Filter video frames: ```wl In[1]:= DifferentiatorFilter[Video["ExampleData/fish.mp4"], 3] Out[1]= \!\(\*VideoBox["![Embedded Video Player](video://content-la04z)"]\) ``` --- Differentiation of a 3D image: ```wl In[1]:= 200DifferentiatorFilter[[image], Pi] Out[1]= [image] ``` --- Filter using exact precision: ```wl In[1]:= DifferentiatorFilter[{0, 0, 0, 1, 1, 1, 0, 0, 0}, Pi, 4] Out[1]= {0, 0, -(8/207 π), (631/207 π), -(8/207 π), (8/207 π), -(631/207 π), (8/207 π), 0} ``` #### Parameters (3) With an audio signal, a numeric cutoff frequency is interpreted as radians per second: ```wl In[1]:= a = \!\(\*AudioBox["![Embedded Audio Player](audio://content-iz4x8)"]\); In[2]:= DifferentiatorFilter[a, 10000 2 Pi] === DifferentiatorFilter[a, Quantity[10000*2*Pi, "Radians"/"Seconds"]] === DifferentiatorFilter[a, Quantity[10000, "Hertz"]] Out[2]= True ``` --- Differentiation of a unit step sequence using a filter of length 7: ```wl In[1]:= DifferentiatorFilter[{0., 0., 0., 0., 1., 1., 1., 1.}, Pi, 6] Out[1]= {0., 0., 0.00442866, -0.0525, 1.10973, -0.0525, 0.00442866, 1.5178830414797062`*^-16} ``` Use a different cutoff frequency: ```wl In[2]:= DifferentiatorFilter[{0., 0., 0., 0., 1., 1., 1., 1.}, Pi / 2, 6] Out[2]= {0., 0., 0.00916597, 0.144268, 0.320632, 0.144268, 0.00916597, 5.204170427930421`*^-17} ``` Use a specific window function: ```wl In[3]:= DifferentiatorFilter[{0., 0., 0., 0., 1., 1., 1., 1.}, Pi, 6, BlackmanWindow] Out[3]= {0., 0., 0., -0.0284032, 1.05287, -0.0284032, 6.522560269672795`*^-16, 6.522560269672795`*^-16} ``` Specify the window function as a numeric list: ```wl In[4]:= DifferentiatorFilter[{0., 0., 0., 0., 1., 1., 1., 1.}, Pi, 6, {1, 1, 1, 1, 1, 1}] Out[4]= {0., 0., 0.0509296, -0.0905415, 1.1827, -0.0905415, 0.0509296, 1.3877787807814457`*^-17} ``` --- Use different cutoff frequencies in each dimension: ```wl In[1]:= DifferentiatorFilter[[image], {Pi / 3, Pi}, 20]//ImageAdjust Out[1]= [image] ``` ### Options (4) #### Padding (2) By default, ``"Fixed"`` padding is used: ```wl In[1]:= DifferentiatorFilter[[image], Pi / 3, 21]//ImageAdjust Out[1]= [image] ``` Use ``Padding -> None`` to eliminate potential image border artifacts: ```wl In[2]:= DifferentiatorFilter[[image], Pi / 3, 21, Padding -> None]//ImageAdjust Out[2]= [image] ``` --- Different padding methods result in different edge effects: ```wl In[1]:= ramp = Range[21] / 21. + RandomReal[.1, 21]; GraphicsRow@(ListLinePlot[{ramp, DifferentiatorFilter[ramp, Pi, 6, Padding -> #]}, ...]& /@ {"Fixed", "Periodic", 0}) Out[1]= [image] ``` #### SampleRate (2) Use a half-band differentiating filter, assuming a normalized sample rate of 1: ```wl In[1]:= DifferentiatorFilter[{0, 0, 0, 1, 1, 1, 0, 0, 0}, π / 2., 5] Out[1]= {0., 0.0217391, 0.194734, 0.194734, 0., -0.194734, -0.194734, -0.0217391, 0.} ``` Assume a sample rate of 3: ```wl In[2]:= DifferentiatorFilter[{0, 0, 0, 1, 1, 1, 0, 0, 0}, 3 π / 2., 5, SampleRate -> 3] Out[2]= {0., 0.0217391, 0.194734, 0.194734, 0., -0.194734, -0.194734, -0.0217391, 0.} ``` --- Apply a half-band differentiator filter to audio sampled at a rate of ``Quantity[44100, "Hertz"]`` : ```wl In[1]:= a = AudioGenerator[Sin[11025 2 π #^2]&]; sr = QuantityMagnitude@AudioSampleRate[a] Out[1]= 44100 In[2]:= DifferentiatorFilter[a, sr π / 2., 21]//Periodogram Out[2]= [image] ``` ### Applications (2) Use a derivative filter to highlight edges in an image: ```wl In[1]:= DifferentiatorFilter[[image], Pi, 6]//ImageAdjust Out[1]= [image] ``` --- Convert a triangle wave to a square wave: ```wl In[1]:= x = Table[TriangleWave[n / 32], {n, 0, 63}]//N; y = DifferentiatorFilter[x, Pi, 6, Padding -> "Periodic"]; In[2]:= GraphicsRow[{ListPlot[x, Filling -> 0, ImageSize -> Small], ListPlot[y, Filling -> 0, ImageSize -> Small]}] Out[2]= [image] ``` ### Properties & Relations (7) Create a differentiator filter using ``LeastSquaresFilterKernel`` and a Hamming window: ```wl In[1]:= n = 5; ω = 1.5; win = Array[HammingWindow, n, {-1 / 2, 1 / 2}]; ker = win LeastSquaresFilterKernel[{"Differentiator", ω}, n] Out[1]= {0.0215281, 0.154205, 0, -0.154205, -0.0215281} In[2]:= ListConvolve[ker, {0, 0, 0, 0, 1, 1, 1, 1}, Ceiling[n / 2], 0] Out[2]= {0., 0., 0.0215281, 0.175734, 0.175734, 0.0215281, -0.0215281, -0.175734} ``` Compare with the result of ``DifferentiatorFilter`` : ```wl In[3]:= DifferentiatorFilter[{0, 0, 0, 0, 1, 1, 1, 1}, ω, n, win, Padding -> 0] Out[3]= {0., 0., 0.0215281, 0.175734, 0.175734, 0.0215281, -0.0215281, -0.175734} ``` --- Impulse response of an odd-length, full-band derivative filter of length 21: ```wl In[1]:= h = DifferentiatorFilter[ArrayPad[{1.}, 10], Pi, 21] Out[1]= {-0.00869565, 0.0121445, -0.0217681, 0.0393059, -0.0670675, 0.108696, -0.171138, 0.270605, -0.456406, 0.977656, 0., -0.977656, 0.456406, -0.270605, 0.171138, -0.108696, 0.0670675, -0.0393059, 0.0217681, -0.0121445, 0.00869565} In[2]:= ListPlot[h, PlotRange -> All, Filling -> 0] Out[2]= [image] ``` Magnitude spectrum of the filter: ```wl In[3]:= Plot[Abs[ListFourierSequenceTransform[h, ω]], {ω, 0, π}] Out[3]= [image] ``` --- Magnitude spectrum of an odd-length, full-band derivative filter with no smoothing window: ```wl In[1]:= Plot[Evaluate@Abs[ListFourierSequenceTransform[DifferentiatorFilter[ArrayPad[{1.}, 10], Pi, 21, None], ω]], {ω, 0, π}] Out[1]= [image] ``` --- Impulse response of an even-length derivative filter: ```wl In[1]:= h = DifferentiatorFilter[ArrayPad[{1.}, 10], Pi, 20] Out[1]= {0., -0.000306694, 0.000492079, -0.0010368, 0.00221336, -0.00453956, 0.00913553, -0.0188871, 0.0434262, -0.133687, 1.26531, -1.26531, 0.133687, -0.0434262, 0.0188871, -0.00913553, 0.00453956, -0.00221336, 0.0010368, -0.000492079, 0.000306694} In[2]:= ListPlot[h, PlotRange -> All, Filling -> 0] Out[2]= [image] ``` Magnitude spectrum of the filter: ```wl In[3]:= Plot[Abs[ListFourierSequenceTransform[h, ω]], {ω, 0, π}, PlotRange -> All] Out[3]= [image] ``` --- Magnitude spectrum of an even-length, full-band derivative filter with no smoothing window: ```wl In[1]:= Plot[Evaluate@Abs[ListFourierSequenceTransform[DifferentiatorFilter[ArrayPad[{1.}, 10], Pi, 20, None], ω]], {ω, 0, π}] Out[1]= [image] ``` --- Magnitude response of an odd-length differentiator filter for different filter lengths: ```wl In[1]:= Manipulate[ ListLinePlot[Abs[Fourier[DifferentiatorFilter[ArrayPad[{1.}, 128], Pi, n], FourierParameters -> {1, -1}]][[ ;; 129]], ...], {{n, 21}, 3, 127, 2}] Out[1]= DynamicModule[«8»] ``` --- Impulse response of a half-band, odd-length derivative filter: ```wl In[1]:= h = DifferentiatorFilter[ArrayPad[{1.}, 10], Pi / 2, 21] Out[1]= {0.00434783, 0.000429523, -0.010884, -0.00178735, 0.0335338, 0.00691978, -0.0855689, -0.0287121, 0.228203, 0.311198, 0., -0.311198, -0.228203, 0.0287121, 0.0855689, -0.00691978, -0.0335338, 0.00178735, 0.010884, -0.000429523, -0.00434783} In[2]:= ListPlot[h, PlotRange -> All, Filling -> 0] Out[2]= [image] ``` Magnitude spectrum of the filter: ```wl In[3]:= Plot[Abs[ListFourierSequenceTransform[h, ω]], {ω, 0, π}] Out[3]= [image] ``` ### Possible Issues (1) With ``Padding -> None``, the output will be shorter than the input: ```wl In[1]:= DifferentiatorFilter[{0., 0., 0., 1., 1., 1.}, 1., Padding -> None] Out[1]= {0.109492} ``` ## See Also * [`LowpassFilter`](https://reference.wolfram.com/language/ref/LowpassFilter.en.md) * [`HilbertFilter`](https://reference.wolfram.com/language/ref/HilbertFilter.en.md) * [`LeastSquaresFilterKernel`](https://reference.wolfram.com/language/ref/LeastSquaresFilterKernel.en.md) * [`ListConvolve`](https://reference.wolfram.com/language/ref/ListConvolve.en.md) * [`ImageConvolve`](https://reference.wolfram.com/language/ref/ImageConvolve.en.md) * [`DerivativeFilter`](https://reference.wolfram.com/language/ref/DerivativeFilter.en.md) * [`GaussianFilter`](https://reference.wolfram.com/language/ref/GaussianFilter.en.md) ## Related Guides * [Linear and Nonlinear Filters](https://reference.wolfram.com/language/guide/LinearAndNonlinearFilters.en.md) * [Image Filtering & Neighborhood Processing](https://reference.wolfram.com/language/guide/ImageFilteringAndNeighborhoodProcessing.en.md) * [Signal Filtering & Filter Design](https://reference.wolfram.com/language/guide/SignalFilteringAndFilterDesign.en.md) * [Video Computation: Update History](https://reference.wolfram.com/language/guide/VideoComputation-UpdateHistory.en.md) ## History * [Introduced in 2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) \| [Updated in 2015 (10.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn102.en.md) ▪ [2016 (11.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn110.en.md) ▪ [2025 (14.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn143.en.md)