Z-evaluator

Z-eval is a single-header C++11 library for efficient transformation and real-time evaluation of Discrete Transfer Functions (DTFs) in the difference-equation form -- you simply have to specify the numerator and denominator coefficients.

Table of Contents

• Overview
  • Making a Discrete TF
  • Using a Discrete TF
  • Algebraic/Utility DTFs
    • Integral / Derivative / Delay
    • Low-Pass Filters
  • Points of Configuration
  • Advanced Features
• API Reference
• Benchmarks
• License

Overview

Transfer functions are an incredibly helpful tool for describing Linear Time Invariant (LTI) systems that make up a key part of Control Theory and Digital Signals Processing.

At its core, Z-eval is a small and efficient utility serving a very narrow purpose -- abstracting away the boilerplate of implementing and evaluating difference equation forms of Discrete Transfer Functions. This utility specifically focuses on resource-limited embedded targets.

For those just starting out in DSP or Control Theory, Z-eval also provides a few helper functions for creating Moving Average and IIR filters, derivative and integral approximations, and group delays without requiring knowledge of any of the underlying arithmetic.

Creating a Discrete TF

This is the general form you follow to create any DTF that you desire:

#include <zeval.hpp>

auto Gz = zeval::makeDTF(
    zeval::makeCs(1.0f),         // Numerator coefficients
    zeval::makeCs(1.0f, -0.5f),  // Denominator coefficients
    TickPolicy{},                // System tick policy object
    100                          // Sampling time (in TickPolicy tick units)
);

You simply specify coefficients as they appear in your discretized TF, along with the sampling time used for the discretization. The only point worth noting is the TickPolicy class that must be defined by the user such that the created zeval::DTF object knows when to compute the next output value in the interval defined by the sampling time argument.

As an example, implementing a TickPolicy class for an Arduino project that wishes to use a time base in milliseconds would simply look like this:

struct TickPolicy {
    static uint32_t getTick() const { return millis(); }
};

Using a Discrete TF

Using the resulting zeval::DTF object Gz is very straightforward -- it's a function object, so you can simply call it as if it were a function:

float input = 42;
float output = Gz(input);

Detail: the default interpolation behavior between sampling periods is the Zero-Order Hold (ZOH), meaning Gz output will remain at its previous value until the subsequent sampling period has elapsed. Specifying custom behavior is in scope for future development.

Algebraic/Utility DTFs

There are a few special numerator and denominator coefficient sets for discrete transfer functions that create some general mathematical operators, such as: derivative, integral, delay, moving average, etc. This library provides a convenient set of utility functions to create these operators along with two very basic low pass filters:

• dtfe::util::makeDerivative()
• dtfe::util::makeIntegral()
• dtfe::util::makeDelay()
• dtfe::util::makeMovingAvg()
• dtfe::util::makeSinglePoleIIR()

Integral / Derivative / Delay

For example, if you need to integrate a signal every 10 ms using the Trapezoidal Rule, then you can simply create a trapezoidal integrator DTF and pass it your signal, like so:

// Size of `Gz_integrator` is only 32 bytes (using `float` for calculations)
auto Gz_integrator = zeval::util::makeTrapezoidalIntegrator(TickPolicy{}, 10);

// Emulating some data acquisition loop
while (is_reading) {
	Gz_integrator(readSignal());
}

// Tip: parameter-less call returns the last computed DTF value
float signal_sum = Gz_integrator(); 

There are several possible methods of integration that you can represent with a DTF, and for some of which Z-eval provides helper constructor functions for:

• zeval::util::makeForwardIntegrator()
• zeval::util::makeBackwardIntegrator()
• zeval::util::makeTrapezoidalIntegrator()

Low-Pass Filters

If you want to smooth out your noisy sensor reading or some other signal, and do no need to or want to design a custom filter, then you're likely looking for one of these two very simple Low-Pass Filters (LPFs):

• Moving Average (FIR)
• Exponential Smoothing (IIR)

These two LPF methods are effective for most simple "signal smoothing" use cases, but they both have their Pros and Cons, some of which are enumerated below:

Filter	Stability	Phase Delay	Roundoff Error	Memory Use	Compute Time
FIR	Guaranteed	Uniform	Constant	Low - High	Low - High
IIR	Not guaranteed	Varying	Accumulates	Lowest	Lowest

Points of Configuration

Zeval is implemented with configurability in mind by using "policy" classes passed as template parameters. Zeval provides a number of various policies for coefficient sets, platform-specific system ticks, and for DTF interpolation methods.

Advanced Features

TODO:

• Compile-time coefficient expansion for N-order delays
• DTF wrappers, e.g. for implementation of true derivative

Benchmarking

TODO:

• Space+time data

API Reference

zeval::makeCs()

template<class... TCs>
constexpr auto makeCs(const TCs&... cs) -> DistinctCSet<...>

This function is used to produce a set of distinct coefficients, used for specifying denominator and numerator TF coefficients.

zeval::makeUniformCs()

template<class... TCs>
constexpr auto makeUniformCs(const TCs&... cs) -> UniformCSet<...>

This function is used to produce a set of coefficients of same value, used for specifying denominator and numerator TF coefficients.

zeval::makeDTF()

template<class TAs,  class TBs,  class TickPolicy>
constexpr auto makeDTF(const TAs& As,  const TBs& Bs, TickPolicy, uint16_t Ts)
    -> DTF<...>

This function creates a callable object, representing the DTF with provided coefficients, that is used to perform the difference equation evaluation

zeval::util::makeMovingAvg()

template<size_t TNum,  class TickPolicy>
constexpr auto makeMovingAvg(TickPolicy p, uint16_t Ts) -> DTF<...>

This is a utility function for quickly generating a moving average (also sometimes called "boxcar" or "FIR") low pass filter DTF object.

zeval::util::makeSinglePoleIIR()

template<class TickPolicy>
constexpr auto makeSinglePoleIIR(float alpha, TickPolicy p, uint16_t Ts) -> DTF<...>

This is a utility function for quickly generating a single-pole infinite impulse response (also sometimes called "recursive") low pass filter DTF object.

zeval::util::makeDerivative()

template<class TickPolicy>
constexpr auto makeDerivative(TTickPolicy p, uint16_t Ts) -> DTF<...>

This is a utility function for quickly generating a derivative approximating DTF object.

zeval::util::makeIntegral()

template<class TickPolicy>
constexpr auto makeIntegral(TTickPolicy p, uint16_t Ts) -> DTF<...>

This is a utility function for quickly generating an integral approximating DTF object.

zeval::util::makeDelay()

template<size_t TNum, class TickPolicy>
constexpr auto makeDelay(TTickPolicy p, uint16_t Ts) -> DTF<...>

This is a utility function for quickly generating a signal-delaying DTF object.

License

Please see the LICENSE file for details.