Implements a generic polyphase FIR filter.

Notes¶

The polyphase feedforward taps \(h_i[n]\) are related to the prototype feedforward taps \(h[n]\), given the number of polyphase branches \(B\), by

The input signal \(x[n]\) can be passed to each polyphase partition (used in interpolation) or commutated from bottom to top (used in decimation).

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

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

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

The output of each polyphase partition can be summed to produce the output signal \(y[n]\) (used in decimation), commutated from top to bottom (used in interpolation), or taken as parallel outputs \(y_i[n]\) (used in channelization).

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

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

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

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

Constructors¶

PolyphaseFIR(branches: int, taps: ArrayLike, ...)

Creates a polyphase FIR 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. The impulse response \(h[n]\) is the filter output when the input is an impulse \(\delta[n]\).

step_response(N: int | None = None) → NDArray

Returns the step response \(s[n]\) of the FIR 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: 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.

Properties¶

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. The delay indicates the output sample index that corresponds to the first input sample.