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An EMA (**Exponential Moving Average**) filter behaves like an RC lowpass filter with RC = SamplePeriod((1-alpha)/alpha) see [here](https://en.wikipedia.org/wiki/Low-pass_filter#Simple_infinite_impulse_response_filter).<br/>
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An EMA (**Exponential Moving Average**) filter behaves like an RC Lowpass filter with RC = SamplePeriod((1-alpha)/alpha) see [here](https://en.wikipedia.org/wiki/Low-pass_filter#Simple_infinite_impulse_response_filter).<br/>
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An EMA filter is implemented by e.g. the following statement:
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```c++
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```c+
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int16_t sLowpass3;
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int16_t Lowpass5;
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...
@@ -70,93 +70,102 @@ The alpha's for the implemented **ultra fast EMA filters** are 1/2, 1/4, 1/8, 1/
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### For a 1 kHz sampling rate (1/1000s sampling interval) we get the following equivalent cutoff (-3db) frequencies:
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Simplified formula for small alpha is CutoffFrequency = (SampleFrequency / (2π * ((1/alpha) - 1));
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- For alpha 1/2 (shift 1) -> (160 Hz / 1) With 1 = (1/alpha) - 1
Using the [more exact formula](https://github.com/MakeMagazinDE/DigitaleFilter/blob/main/Filter-Berechnungen.xlsx), alpha = 1 - e ^ -((2π * CutoffFrequency) / SampleFrequency)
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=> CutoffFrequency = (-ln(1-alpha) * SampleFrequency) / 2π we get the values:
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- 110 Hz
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- 46 Hz
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- 21.2 Hz
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- 10.3 Hz
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- 5.05 Hz
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-1/2 -> 110 Hz
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-1/4 -> 46 Hz
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-1/8 -> 21.2 Hz
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-1/16 -> 10.3 Hz
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-1/32 -> 5.05 Hz
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The **maximum output value for integer filters** is: InputValue - ((1 << (ShiftValue -1)) -1), i.e **85 for InputValue 100 and ShiftValue 5** (Lowpass5).
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### The [SimpleEMAFilters.hpp](https://github.com/ArminJo/Arduino-Utils/blob/master/src/SimpleEMAFilters.hpp#L66) contains:
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- A set of **ultrafast** EMA (Exponential Moving Average) filters which require only **1 to 2 microseconds**.
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- 3 highpass and bandpass filters, generated by just subtracting one lowpass from input (highpass) or from another lowpass (bandpass).
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- 16 and 32 bit Biquad filters.
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- 3 Highpass and Bandpass filters, generated by just subtracting one Lowpass from input (Highpass) or from another Lowpass (Bandpass).
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- 16 and 32 bit Biquad filters / State Variable Filter (SVF).
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- Display routines for Arduino Plotter.
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All implemented filters are applied at once to the input test signal calling `doFiltersStep(int16_t aInput)` and the results can in turn easily be displayed in the Arduino Plotter.
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### Plotter outputs representing e.g. a 42, 21, 10.5, 5.2 Hz square / sine wave at a sample rate of 1 ms (or 84, 42 ... Hz at a sample rate of 0.5 ms and so on)
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### Arduino 1.8.19 plotter outputs are showing a 42, 21, 10.5, 5.2 Hz square / sine wave at a sample rate of 1 ms (or 84, 42 ... Hz at a sample rate of 0.5 ms and so on)
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The outputs are generated by the EMAFilterDemo example of this library.
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Arduino Plotter output for a rectangle input signal with a amplitude of +/- 100. E.g. LowP3 is the Lowpass with alpha 1/8 implemented by `>> 3` and the cutoff frequency of 21 Hz.
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Output for a rectangle input signal with a amplitude of +/- 100.
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E.g. LowP3 is the Lowpass implemented by `>> 3` corresponding to alpha = 1/8 with a cutoff frequency of 21 Hz.<br/>
-Therefore highpass -here HighP3_16- can have an overshoot of 100%.
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- The output of DoubleLowpass3 and 4 is more and more resembling a delayed sine.
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- Highpass is always Input - Lowpass, therefore Highpass -here HighP3_16- can have an overshoot of up to 100%.
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-BandStop is always Input - Bandpass.
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- The output of DoubleLowpass3 and 4 resembes a delayed sine the higher the frequency is.
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- The output of LowP5_32 and DoubleLowP4_16 have almost the same amplitude but DoubleLowP4_16 is way more resembling a (delayed) sine.
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- The output of TripleLowP3_16 has a higher amplitude than DoubleLowP4_16 with both looking quite like a (delayed) sine.
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- The output of Lowpass8 behave like a integrator here, resulting in a triangle output, because it has a very low cutoff frequency.
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- The output of BiQuad tends to oscillate with its resonance frequency.
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<br/>
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Arduino Plotter output for a rectangle input signal with very low amplitude of +/- 20 where you see clipping effects due to the limited resolution of the used 16 bit math.
The Lowpass3_int32 and Lowpass5_int32 are 32 bit fixed point implementations for higher resolution which requires **4.2 µs instead of the 1.8 / 2.5µs of 16 bit**.
The Lowpass3_int32 and Lowpass5_int32 are 32 bit fixed point implementations for higher resolution which requires **3.06 / 3.94 µs instead of the 2.0 / 2.63µs of 16 bit**.
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<br/>
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Arduino Plotter output for a sine input signal with a amplitude of +/- 100.
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Note the **different attenuations** at different frequencies.
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Note the **different attenuations** at different frequencies and the **different phase shifts** of the filters.
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<br/><br/>
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Arduino Plotter output for a triangle input signal with a amplitude of +/- 100.
| Bandpass and reject filters.<br/>Bandpass is generated by subtracting one lowpass from onother with different shift value.<br/>Reject filter is input minus bandpass.| High order Lowpass. |
| LowPass1, 3, 5 and 8.<br/>Lowpass8 behave like a integrator here, resulting in a triangle output. | LowPass3, 5 and 8.<br/>Comparison of different implementations with 16 bit integer, 32 bit integer and 32 bit float. |
| BiQuad filters with damping factor = 0.<br/>You see the overshoot. ||
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|||
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|||
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| Bandpass and Bandstop filters.<br/>Bandpass is generated by subtracting one Lowpass from another with different shift value.<br/>Bandstop filter is input minus Bandpass.| Selected Lowpass.<br/>Lowpass8 behave like a integrator here, resulting in a triangle output. |
Lowpass5_int32 += ((((int32_t) InputValue) << 8) - Lowpass5_int32) >> 5; // Fixed point 4.0 us, value is Lowpass5_int32 >> 8
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Lowpass8_int32 += ((((int32_t) InputValue) << 16) - Lowpass8_int32) >> 8; // Fixed point 2.13 us because of fast 8 bit shift, value is Lowpass8_int32 >> 16
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```
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**!Attention!** The 16 bit implementations are limited to a **maximum input value of +/- 16383** for rectangular input (which is the worst input case). The reason is, that the term `InputValue - Lowpass3` must always fit into a 16 bit signed integer.
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