Laplace random number，laplace transform！

# NWaves [](https://opensource.org/licenses/MIT)   [](https://gitter.im/NWaves/community?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge)  NWaves is a .NET DSP library with a lot of audio processing functions. ## Releases NWaves is [available on NuGet](https://www.nuget.org/packages/NWaves/): ```PM> Install-Package NWaves``` [Read wiki documentation](https://github.com/ar1st0crat/NWaves/wiki) New version **0.9.6** is out! Faster, smarter, more features. [Read about changes here](https://github.com/ar1st0crat/NWaves/wiki/Known-bugs-and-changelog) [Notes for non-experts in DSP](https://github.com/ar1st0crat/NWaves/wiki/Notes-for-non~experts-in-DSP) [NWaves for MATLAB/sciPy users](https://github.com/ar1st0crat/NWaves/wiki/NWaves-for-MATLAB-and-sciPy-users) [Watch survey video](https://www.youtube.com/watch?v=GyRixqQ613A) | [Playlist](https://www.youtube.com/playlist?list=PL1BjzwHbvgSWVbr32uA2ihdnVCLhVA8a5) | [Samples](https://github.com/ar1st0crat/NWaves.Samples) | [Benchmarks](https://github.com/ar1st0crat/NWaves/tree/master/NWaves.Benchmarks) | [Playground](https://ar1st0crat.github.io/NWaves.Playground/) ([code](https://github.com/ar1st0crat/NWaves.Playground)) ## Main features - [x] major DSP transforms (FFT, DCT, MDCT, STFT, FWT, Hilbert, Hartley, Mellin, cepstral, Goertzel) - [x] signal builders (sine, white/pink/red/Perlin noise, awgn, triangle, sawtooth, square, pulse, ramp, ADSR, wavetable) - [x] basic LTI digital filters (moving average, comb, Savitzky-Golay, pre/de-emphasis, DC removal, RASTA) - [x] FIR/IIR filtering (offline and online), zero-phase filtering - [x] BiQuad filters (low-pass, high-pass, band-pass, notch, all-pass, peaking, shelving) - [x] 1-pole filters (low-pass, high-pass) - [x] IIR filters (Bessel, Butterworth, Chebyshev I & II, Elliptic, Thiran) - [x] basic operations (convolution, cross-correlation, rectification, amplification, fade / crossfade) - [x] block convolution (overlap-add / overlap-save offline and online) - [x] basic filter design & analysis (group delay, zeros/poles, BP, BR, HP from/to LP, SOS, combining filters) - [x] state space representation of LTI filters - [x] FIR filter design: frequency sampling, window-sinc, equiripple (Remez / Parks-McClellan) - [x] IIR filter design: IirNotch / IirPeak / IirCombNotch / IirCombPeak - [x] non-linear filters (median filter, distortion effects, bit crusher) - [x] windowing functions (Hamming, Blackman, Hann, Gaussian, Kaiser, KBD, triangular, Lanczos, flat-top, Bartlett) - [x] periodograms (Welch / Lomb-Scargle) - [x] psychoacoustic filter banks (Mel, Bark, Critical Bands, ERB, octaves) and VTLN warping - [x] customizable feature extraction (time-domain, spectral, MFCC, PNCC/SPNCC, LPC, LPCC, PLP, AMS) - [x] preconfigured MFCC extractors: HTK (MFCC-FB24), Slaney (MFCC-FB40) - [x] LPC conversions: LPC<->cepstrum, LPC<->LSF - [x] feature post-processing (mean and variance normalization, adding deltas) and CSV serialization - [x] spectral features (centroid, spread, flatness, entropy, rolloff, contrast, crest, decrease, noiseness, MPEG7) - [x] harmonic features (harmonic centroid and spread, inharmonicity, tristimulus, odd-to-even ratio) - [x] time-domain characteristics (rms, energy, zero-crossing rate, entropy) - [x] pitch tracking (autocorrelation, YIN, ZCR + Schmitt trigger, HSS/HPS, cepstrum) - [x] chromagram (chroma feature extractor) - [x] time scale modification (phase vocoder, PV with identity phase locking, WSOLA, PaulStretch) - [x] simple resampling, interpolation, decimation - [x] bandlimited resampling - [x] wavelets: haar, db, symlet, coiflet - [x] polyphase filters - [x] noise reduction (spectral subtraction, sciPy-style Wiener filtering) - [x] sound effects (echo, tremolo, wahwah, phaser, chorus, vibrato, flanger, pitch shift, morphing, robotize, whisperize) - [x] 3D/Stereo audio (stereo panning, stereo and ping-pong delay, ITD-ILD, binaural panning) - [x] envelope following - [x] dynamics processing (limiter / compressor / expander / noise gate) - [x] harmonic/percussive separation - [x] Griffin-Lim algorithm - [x] Karplus-Strong synthesis - [x] PADSynth synthesis - [x] adaptive filtering (LMS, NLMS, LMF, SignLMS, RLS) - [x] simple modulation/demodulation (AM, ring, FM, PM) - [x] simple audio playback and recording ## Philosophy of NWaves NWaves was initially intended for research, visualizing and teaching basics of DSP and sound programming. Usually, DSP code is quite complicated and difficult to read, because it's full of optimizations (which is actually a very good thing). NWaves project aims in particular at achieving a tradeoff between good understandable code/design and satisfactory performance. Yet, the main purpose of this lib is to offer the DSP codebase that would be: - easy to read and understand - easy to incorporate into existing projects - easy to port to other programming languages and frameworks - even possibly treated as the DSP/audio textbook. According to NWaves architecture, there are following general reusable building blocks for all kinds of DSP tasks: [Transforms](#transforms) | [Filters](#filters-and-effects) | [Signal builders](#signal-builders) | [Feature extractors](#feature-extractors) ## Quickstart ### Working with 1D signals ```C# // Create signal from samples repeated 100 times float[] samples = new [] { 0.5f, 0.2f, -0.3f, 1.2f, 1.6f, -1.8f, 0.3f, -0.2f }; var s = new DiscreteSignal(8000, samples).Repeat(100); var length = s.Length; var duration = s.Duration; var echoSignal = s + s.Delay(50); var marginSignal = s.First(64).Concatenate(s.Last(64)); var repeatMiddle = s[400, 500].Repeat(10); var mean = s.Samples.Average(); var sigma = s.Samples.Average(x => (x - mean) * (x - mean)); var normSignal = s - mean; normSignal.Attenuate(sigma); ``` ### Signal builders ```C# DiscreteSignal sinusoid = new SineBuilder() .SetParameter("frequency", 500.0/*Hz*/) .SetParameter("phase", Math.PI / 6) .OfLength(1000) .SampledAt(44100/*Hz*/) .Build(); DiscreteSignal noise = new RedNoiseBuilder() .SetParameter("min", -2.5) .SetParameter("max", 2.5) .OfLength(800) .SampledAt(44100) .DelayedBy(200) .Build(); DiscreteSignal noisy = new SineBuilder() .SetParameter("min", -10.0) .SetParameter("max", 10.0) .SetParameter("freq", 1200.0/*Hz*/) .OfLength(1000) .SampledAt(44100) .SuperimposedWith(noise) .Build(); ``` Signal builders can also act as real-time generators of samples: ```C# SignalBuilder lfo = new TriangleWaveBuilder() .SetParameter("min", 100) .SetParameter("max", 1500) .SetParameter("frequency", 2.0/*Hz*/) .SampledAt(16000/*Hz*/); //while (...) { var sample = lfo.NextSample(); //... } ``` ### Signals and wave files: ```C# WaveFile waveContainer; // load using (var stream = new FileStream("sample.wav", FileMode.Open)) { waveContainer = new WaveFile(stream); } DiscreteSignal left = waveContainer[Channels.Left]; DiscreteSignal right = waveContainer[Channels.Right]; // save var waveFileOut = new WaveFile(left); using (var stream = new FileStream("saved_mono.wav", FileMode.Create)) { waveFileOut.SaveTo(stream); } var waveFileStereo = new WaveFile(new [] { left, right }); using (var stream = new FileStream("saved_stereo.wav", FileMode.Create)) { waveFileStereo.SaveTo(stream); } ``` ### Transforms For each transform there's a corresponding transformer object. Each transformer object has ```Direct()``` and ```I