sdr.Integrator

class sdr.Integrator(sdr.IIR)

Implements an integrator IIR filter.

Notes¶

A discrete-time integrator is an IIR filter that continuously accumulates the input signal. Accordingly, it has infinite gain at DC.

The backward integrator is defined by:

y [n] = y [n - 1] + x [n - 1]

H (z) = \frac{z^{- 1}}{1 - z^{- 1}}

The trapezoidal integrator is defined by:

y [n] = y [n - 1] + \frac{1}{2} x [n] + \frac{1}{2} x [n - 1]

H (z) = \frac{1}{2} \frac{1 + z^{- 1}}{1 - z^{- 1}}

The forward integrator is defined by:

y [n] = y [n - 1] + x [n]

H (z) = \frac{1}{1 - z^{- 1}}

Examples¶

Create integrating IIR filters.

In [1]: iir_back = sdr.Integrator("backward"); \
   ...: iir_trap = sdr.Integrator("trapezoidal"); \
   ...: iir_forw = sdr.Integrator("forward")
   ...: 

Integrate a Gaussian pulse.

In [2]: x = sdr.gaussian(0.3, 5, 10); \
   ...: y_back = iir_back(x); \
   ...: y_trap = iir_trap(x); \
   ...: y_forw = iir_forw(x)
   ...: 

In [3]: plt.figure(); \
   ...: sdr.plot.time_domain(x, label="Input"); \
   ...: sdr.plot.time_domain(y_back, label="Integral (backward)"); \
   ...: sdr.plot.time_domain(y_trap, label="Integral (trapezoidal)"); \
   ...: sdr.plot.time_domain(y_forw, label="Integral (forward)"); \
   ...: plt.title("Discrete-time integration of a Gaussian pulse");
   ...: 

Integrate a raised cosine pulse.

In [4]: x = sdr.root_raised_cosine(0.1, 8, 10); \
   ...: y_back = iir_back(x); \
   ...: y_trap = iir_trap(x); \
   ...: y_forw = iir_forw(x)
   ...: 

In [5]: plt.figure(); \
   ...: sdr.plot.time_domain(x, label="Input"); \
   ...: sdr.plot.time_domain(y_back, label="Integral (backward)"); \
   ...: sdr.plot.time_domain(y_trap, label="Integral (trapezoidal)"); \
   ...: sdr.plot.time_domain(y_forw, label="Integral (forward)"); \
   ...: plt.title("Discrete-time integration of a raised cosine pulse");
   ...: 

Plot the frequency responses.

In [6]: plt.figure(); \
   ...: sdr.plot.magnitude_response(iir_back, label="Backward"); \
   ...: sdr.plot.magnitude_response(iir_trap, label="Trapezoidal"); \
   ...: sdr.plot.magnitude_response(iir_forw, label="Forward"); \
   ...: f = np.linspace(0, 0.5, 100); \
   ...: plt.plot(f, sdr.db(np.abs(1/(2 * np.pi * f))**2), color="k", linestyle="--", label="Theory"); \
   ...: plt.legend(); \
   ...: plt.title("Magnitude response of integrating IIR filters");
   ...: 

Constructors¶

Integrator(...): Creates an integrating 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. The impulse response $h [n]$ is the filter output when the input is an impulse $δ [n]$ .

step_response(N: int = 100) → NDArray: Returns the step response $s [n]$ of the IIR filter. The step response $s [n]$ is the filter output when the input is a unit step $u [n]$ .

frequency_response(...) → tuple[numpy.ndarray[Any, numpy.dtype[numpy.float64]], numpy.ndarray[Any, numpy.dtype[numpy.complex128]]]
frequency_response(freqs, ...) → complex
frequency_response(freqs, ...) → ndarray[Any, dtype[complex128]]: Returns the frequency response $H (ω)$ 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.