sdr.Differentiator

class sdr.Differentiator(sdr.FIR)

Implements a differentiator FIR filter.

Notes¶

A discrete-time differentiator is an FIR filter with impulse response

h [n] = \frac{(- 1)^{n}}{n} \cdot h_{w i n} [n], - \frac{N}{2} \leq n \leq \frac{N}{2} .

The truncated impulse response is multiplied by the windowing function $h_{w i n} [n]$ .

References¶

Michael Rice, Digital Communications: A Discrete Time Approach, Section 3.3.3.

Examples¶

Create a differentiator FIR filter.

In [1]: fir = sdr.Differentiator()

Differentiate a Gaussian pulse.

In [2]: x = sdr.gaussian(0.3, 5, 10); \
   ...: y = fir(x, "same")
   ...: 

In [3]: plt.figure(); \
   ...: sdr.plot.time_domain(x, label="Input"); \
   ...: sdr.plot.time_domain(y, label="Derivative"); \
   ...: plt.title("Discrete-time differentiation of a Gaussian pulse");
   ...: 

Differentiate a raised cosine pulse.

In [4]: x = sdr.root_raised_cosine(0.1, 8, 10); \
   ...: y = fir(x, "same")
   ...: 

In [5]: plt.figure(); \
   ...: sdr.plot.time_domain(x, label="Input"); \
   ...: sdr.plot.time_domain(y, label="Derivative"); \
   ...: plt.title("Discrete-time differentiation of a raised cosine pulse");
   ...: 

Plot the frequency response across filter order.

In [6]: fir_2 = sdr.Differentiator(2); \
   ...: fir_6 = sdr.Differentiator(6); \
   ...: fir_10 = sdr.Differentiator(10); \
   ...: fir_20 = sdr.Differentiator(20); \
   ...: fir_40 = sdr.Differentiator(40); \
   ...: fir_80 = sdr.Differentiator(80)
   ...: 

In [7]: plt.figure(); \
   ...: sdr.plot.magnitude_response(fir_2, y_axis="linear", label="N=2"); \
   ...: sdr.plot.magnitude_response(fir_6, y_axis="linear", label="N=6"); \
   ...: sdr.plot.magnitude_response(fir_10, y_axis="linear", label="N=10"); \
   ...: sdr.plot.magnitude_response(fir_20, y_axis="linear", label="N=20"); \
   ...: sdr.plot.magnitude_response(fir_40, y_axis="linear", label="N=40"); \
   ...: sdr.plot.magnitude_response(fir_80, y_axis="linear", label="N=80"); \
   ...: f = np.linspace(0, 0.5, 100); \
   ...: plt.plot(f, np.abs(2 * np.pi * f)**2, color="k", linestyle="--", label="Theory"); \
   ...: plt.legend(); \
   ...: plt.title("Magnitude response of differentiator FIR filters");
   ...: 

Constructors¶

Differentiator(order: int = 20, ...): Creates a differentiator FIR filter.

Special methods¶

__call__(x: ArrayLike, ...) → NDArray: Filters the input signal $x [n]$ with the FIR filter.

__len__() → int: Returns the filter length $N + 1$ .

Streaming mode only¶

reset(): Resets the filter state. Only useful when using streaming mode.

flush() → NDArray: Flushes the filter state by passing zeros through the filter. Only useful when using streaming mode.

property streaming : bool: Indicates whether the filter is in streaming mode.

property state : NDArray: The filter state consisting of the previous $N$ inputs.

Methods¶

impulse_response(n: int | None = None) → NDArray: Returns the impulse response $h [n]$ of the FIR filter.

step_response(n: int | None = None) → NDArray: Returns the step response $s [n]$ of the FIR filter.

frequency_response(...) → tuple[ndarray[tuple[int, ...], dtype[float64]], ndarray[tuple[int, ...], dtype[complex128]]]
frequency_response(freqs: float, ...) → complex
frequency_response(...) → ndarray[tuple[int, ...], dtype[complex128]]: Returns the frequency response $H (ω)$ of the FIR filter.

group_delay(...) → tuple[TypeAliasForwardRef('~numpy.typing.NDArray'), TypeAliasForwardRef('~numpy.typing.NDArray')]: Returns the group delay $τ_{g} (ω)$ of the FIR filter.

phase_delay(...) → tuple[TypeAliasForwardRef('~numpy.typing.NDArray'), TypeAliasForwardRef('~numpy.typing.NDArray')]: Returns the phase delay $τ_{ϕ} (ω)$ of the FIR filter.

noise_bandwidth(sample_rate: float = 1.0) → float: Returns the noise bandwidth $B_{n}$ of the FIR filter.

Properties¶

property taps : NDArray: The feedforward taps $h [n]$ with length $N + 1$ .

property order : int: The order of the FIR filter $N$ .

property delay : int: The delay of the FIR filter $d = ⌊ \frac{N + 1}{2} ⌋$ in samples.