Implements a polyphase decimating FIR filter.

Notes

The polyphase decimating filter is equivalent to first filtering the input signal \(x[n]\) with the prototype FIR filter with feedforward coefficients \(h[n]\) and then downsampling the filtered signal by \(Q\) (by discarding \(Q-1\) samples every \(Q\) samples).

Instead, the polyphase decimating filter first decomposes the prototype FIR filter into \(Q\) polyphase filters with feedforward coefficients \(h_i[n]\). The polyphase filters are then applied to the commutated input signal \(x[n]\) in parallel. The outputs of the polyphase filters are then summed. This prevents the need to compute outputs that will be discarded, as is done in the first case.

                          +------------------------+
 ..., x[6], x[3], x[0] -->| h[0], h[3], h[6], h[9] |---+
                          +------------------------+   |
                          +------------------------+   v
 ..., x[5], x[2], 0    -->| h[1], h[4], h[7], 0    |-->@--> ..., y[1], y[0]
                          +------------------------+   ^
                          +------------------------+   |
 ..., x[4], x[1], 0    -->| h[2], h[5], h[8], 0    |---+
                          +------------------------+

 Input Commutator                                           Output Summation
 (bottom-to-top)

 x[n] = Input signal with sample rate fs
 y[n] = Output signal with sample rate fs / Q
 h[n] = Prototype FIR filter
 @ = Adder

The polyphase feedforward taps \(h_i[n]\) are related to the prototype feedforward taps \(h[n]\) by

References

• fred harris, Multirate Signal Processing for Communication Systems, Chapter 7: Resampling Filters.

Examples

Create an input signal to interpolate.

In [1]: x = np.cos(np.pi / 64 * np.arange(280))

Create a polyphase filter that decimates by 7 using the Kaiser window method.

In [2]: fir = sdr.Decimator(7); fir
Out[2]: sdr.Decimator(7, 'kaiser', streaming=False)

In [3]: y = fir(x)

In [4]: plt.figure(); \
   ...: sdr.plot.time_domain(x, marker=".", label="Input"); \
   ...: sdr.plot.time_domain(y, sample_rate=fir.rate, marker="o", label="Decimated"); \
   ...: plt.title("Decimation by 7 with the Kaiser window method");
   ...: 

Create a streaming polyphase filter that decimates by 7 using the Kaiser window method. This filter preserves state between calls.

In [5]: fir = sdr.Decimator(7, streaming=True); fir
Out[5]: sdr.Decimator(7, 'kaiser', streaming=True)

In [6]: y1 = fir(x[0:70]); \
   ...: y2 = fir(x[70:140]); \
   ...: y3 = fir(x[140:210]); \
   ...: y4 = fir(x[210:280]); \
   ...: y5 = fir.flush()
   ...: 

In [7]: plt.figure(); \
   ...: sdr.plot.time_domain(x, marker=".", label="Input"); \
   ...: sdr.plot.time_domain(y1, sample_rate=fir.rate, offset=-fir.delay/fir.rate + 0, marker="o", label="Decimated $y_1[n]$"); \
   ...: sdr.plot.time_domain(y2, sample_rate=fir.rate, offset=-fir.delay/fir.rate + 70, marker="o", label="Decimated $y_2[n]$"); \
   ...: sdr.plot.time_domain(y3, sample_rate=fir.rate, offset=-fir.delay/fir.rate + 140, marker="o", label="Decimated $y_3[n]$"); \
   ...: sdr.plot.time_domain(y4, sample_rate=fir.rate, offset=-fir.delay/fir.rate + 210, marker="o", label="Decimated $y_4[n]$"); \
   ...: sdr.plot.time_domain(y5, sample_rate=fir.rate, offset=-fir.delay/fir.rate + 280, marker="o", label="Decimated $y_5[n]$"); \
   ...: plt.title("Streaming decimation by 7 with the Kaiser window method");
   ...: 

Constructors¶

Decimator(decimation: int, ...)

Creates a polyphase FIR decimating filter.

Special methods¶

__call__(x: ArrayLike, mode: 'rate' | 'full' = 'rate') → NDArray

Filters the input signal \(x[n]\) with the polyphase 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[Any, dtype[float64]], ndarray[Any, dtype[complex128]]]

frequency_response(freqs: float, ...) → complex

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

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

group_delay(...) → tuple[NDArray, NDArray]

Returns the group delay \(\tau_g(\omega)\) of the FIR filter.

phase_delay(...) → tuple[NDArray, NDArray]

Returns the phase delay \(\tau_{\phi}(\omega)\) of the FIR filter.

noise_bandwidth(sample_rate: float = 1.0) → float

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

Properties¶

property method : 'kaiser' | 'custom'

The method used to design the polyphase decimating filter.

property branches : int

The number of polyphase branches \(B\).

property taps : NDArray

The prototype feedforward taps \(h[n]\).

property polyphase_taps : NDArray

The polyphase feedforward taps \(h_i[n]\).

property order : int

The order \(N = (M + 1)B - 1\) of the FIR prototype filter \(h[n]\).

property polyphase_order : int

The order \(M = (N + 1)/B - 1\) of each FIR polyphase filter \(h_i[n]\).

property input : 'hold' | 'top-to-bottom' | 'bottom-to-top'

The input connection method.

property output : 'sum' | 'top-to-bottom' | 'bottom-to-top' | 'all'

The output connection method.

property interpolation : int

The integer interpolation rate \(P\).

property decimation : int

The integer decimation rate \(Q\).

property rate : float

The fractional resampling rate \(r = P/Q\). The output sample rate is \(f_{s,out} = f_{s,in} \cdot r\).

property delay : int

The delay of polyphase FIR filter in samples.