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@@ -55,17 +55,17 @@ However, after bandlimiting, all harmonics above $f_{sr}/2$ become zero, so its |
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The curve produced by a bandlimited discontinuity is known as the [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon). |
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A DSP algorithm attempting to model a jump found in sawtooth or square waves must include this effect, such as by inserting a minBLEP or polyBLEP signal for each discontinuity. |
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Otherwise higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below $f_{sr}/2$. |
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Otherwise, higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below $f_{sr}/2$. |
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Even signals containing no discontinuities, such as a triangle wave with harmonic amplitudes $(-1)^k / k^2$, must be correctly bandlimited or aliasing will occur. |
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One possible method is to realize that a triangle wave is an integrated square wave, and an integrator is just a filter with a -20dB per [decade](https://en.wikipedia.org/wiki/Decade_(log_scale)) slope. |
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One possible method is to realize that a triangle wave is an integrated square wave, and an integrator is just a filter with a -20dB per [decade](https://en.wikipedia.org/wiki/Decade_%28log_scale%29) slope. |
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Since linear filters commute, a bandlimited integrated square wave is just an integrated bandlimited square wave. |
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The most general approach is to generate samples at a high sample rate, apply a FIR or polyphase filter, and downsample by an integer factor (known as decimation). |
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For more specific applications, more advances techniques exist for certain cases. |
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Aliasing is required for many processes, including waveform generation, waveshaping, distortion, saturation, and typically all nonlinear processes. |
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It is sometimes *not* required for reverb, linear filters, audio-rate FM of sine signals (which is why primitive digital chips in the 80's were able to sound reasonably good), mixing signals, and most other linear processes. |
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Anti-aliasing is required for many processes, including waveform generation, waveshaping, distortion, saturation, and typically all nonlinear processes. |
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It is sometimes *not* required for reverb, linear filters, audio-rate FM of sine signals (which is why primitive digital chips in the 80's were able to sound reasonably good), adding signals, and most other linear processes. |
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### Linear filters |
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@@ -116,13 +116,13 @@ Note that the above formula is the convolution between vectors $y$ and $b$, and |
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$$y \ast b = \mathcal{F}^{-1} \{ \mathcal{F}\{y\} \cdot \mathcal{F}\{b\} \}$$ |
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where $\cdot$ is element-wise multiplication. |
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While the naive FIR formula above is $O(n^2)$ when processing blocks of $n$ samples, the FFT FIR method is $O(\log n)$. |
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While the naive FIR formula above is $O(N^2)$ when processing blocks of $N$ samples, the FFT FIR method is $O(\log N)$. |
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A disadvantage of the FFT FIR method is that the signal must be delayed by $N$ samples to produce any output. |
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You can combine the naive and FFT methods into a hybrid approach with the [overlap-add](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) or [overlap-save](https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods. |
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#### IIR filters |
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An infinite impulse response (IIR) filter is a general rational transfer function. Applying the $H(z)$ operator to an input and output signal, |
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An infinite impulse response (IIR) filter is a general rational transfer function. By multiplying the denominator of the rational $H(z)$ definition above on both sides and applying it to an input and output signal, |
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$$\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}$$ |
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Usually $a_0$ is normalized to 1, and $y_k$ can be written explicitly. |
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$$y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1} a_m y_{k-m}$$ |
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Andrew Belt