class sdr.LeakyIntegrator(sdr.IIR)

Implements a leaky integrator IIR filter.

Notes

A discrete-time leaky integrator is an IIR filter that approximates an FIR moving average. The previous output is remembered with the leaky factor \(\alpha\) and the new input is scaled with \(1 - \alpha\).

The difference equation is

\[y[n] = \alpha \cdot y[n-1] + (1 - \alpha) \cdot x[n] .\]

The transfer functions is

\[H(z) = \frac{1 - \alpha}{1 - \alpha z^{-1}} .\]

IIR Integrator Block Diagram
      1 - α
x[n] ------->@---------------+--> y[n]
             ^               |
           α |   +------+    |
             +---| z^-1 |<---+
                 +------+

Examples

Create an FIR moving average filter and an IIR leaky integrator filter.

In [1]: fir = sdr.MovingAverager(30)

In [2]: iir = sdr.LeakyIntegrator(1 - 2 / 30)

Compare the step responses.

In [3]: plt.figure(); \
   ...: sdr.plot.step_response(fir, N=100, label="Moving Averager"); \
   ...: sdr.plot.step_response(iir, N=100, label="Leaky Integrator");
   ...:
../../_images/sdr_LeakyIntegrator_1.png

Compare the magnitude responses.

In [4]: plt.figure(); \
   ...: sdr.plot.magnitude_response(fir, label="Moving Averager"); \
   ...: sdr.plot.magnitude_response(iir, label="Leaky Integrator"); \
   ...: plt.ylim(-35, 5);
   ...:
../../_images/sdr_LeakyIntegrator_2.png

Compare the output of the two filters to a Gaussian random process.

In [5]: x = np.random.randn(1_000) + 2.0; \
   ...: y_fir = fir(x); \
   ...: y_iir = iir(x)
   ...: 

In [6]: plt.figure(); \
   ...: sdr.plot.time_domain(y_fir, label="Moving Averager"); \
   ...: sdr.plot.time_domain(y_iir, label="Leaky Integrator");
   ...:
../../_images/sdr_LeakyIntegrator_3.png

Constructors

LeakyIntegrator(alpha: float, streaming: bool = False)

Creates a leaky integrator IIR filter.

classmethod ZerosPoles(zeros: ArrayLike, poles, ...) → Self

Creates an IIR filter from its zeros, poles, and gain.

Special methods

__call__(x: ArrayLike) → NDArray

Filters the input signal \(x[n]\) with the IIR filter.

Streaming mode only

reset()

Resets the filter state. Only useful when using streaming mode.

property streaming : bool

Indicates whether the filter is in streaming mode.

property state : NDArray

The filter state.

Methods

impulse_response(N: int = 100) → NDArray

Returns the impulse response \(h[n]\) of the IIR filter.

step_response(N: int = 100) → NDArray

Returns the step response \(s[n]\) of the IIR filter.

frequency_response(...) → tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[complex128]]]
frequency_response(freqs, ...) → complex
frequency_response(freqs, ...) → ndarray[Any, dtype[complex128]]

Returns the frequency response \(H(\omega)\) of the IIR filter.

noise_bandwidth(sample_rate: float = 1.0) → float

Returns the noise bandwidth \(B_n\) of the IIR filter.

Properties

property b_taps : NDArray

The feedforward taps \(b_i\) for \(i = 0,...,M\).

property a_taps : NDArray

The feedback taps \(a_j\) for \(j = 0,...,N\).

property order : int

The order of the IIR filter \(N\).

property zeros : NDArray

The zeros of the IIR filter.

property poles : NDArray

The poles of the IIR filter.

property gain : float

The gain of the IIR filter.