Farrow arbitrary resampler¶

import numpy as np
import matplotlib.pyplot as plt

import sdr

%config InlineBackend.print_figure_kwargs = {"facecolor" : "w"}
%matplotlib inline
# %matplotlib widget

sdr.plot.use_style()

Construct an input signal, $x[n]=x(n{T}_s)$¶

Create a discrete-time signal $x[n]$ that sample the continuous-time signal $x(t)$ at a sample rate $f_s=1/T_s$.

sample_rate = 1  # samples/s
N = 100  # samples
freq = 0.05  # Hz
tx = np.arange(N) / sample_rate  # Time axis for the input signal
x = np.exp(1j * 2 * np.pi * freq * tx)  # Complex exponential input signal
x *= np.exp(-np.arange(N) / 100)  # Exponential decay

plt.figure()
sdr.plot.time_domain(tx, x, marker="o", fillstyle="none")
plt.title("Original signal, $x(t)$")
plt.show()

Resample the input signal with rate $r$, $y[n]=x(n\frac{T_s}{r})$¶

Now, resample $x[n]$ such that the output $y[n]$ is equivalent to sampling the continuous-time $x(t)$ at sample rate $f_s=r/T_s$. This is equivalent to sampling $x[n]$ at $1/r$ fractional samples. This is accomplished, for arbitrary $r$, with a Farrow arbitrary resampler.

In the sdr library, the Farrow arbitrary resampler is implemented in sdr.FarrowResampler.

def resample_signal(rate):
    farrow = sdr.FarrowResampler()
    y = farrow(x, rate)
    new_sample_rate = rate * sample_rate
    ty = np.arange(y.size) / new_sample_rate  # Time axis for output signal

    print(f"Input signal length: {x.size}")
    print(f"Output signal length: {y.size}")

    plt.figure()
    sdr.plot.time_domain(tx, x, linestyle="none", marker="o", fillstyle="none", label="Input")
    plt.gca().set_prop_cycle(None)
    sdr.plot.time_domain(ty, y, linestyle="none", marker=".", label="Output")
    plt.title(f"Original $x(t)$ and resampled signal $y(t)$, rate = {rate}")
    plt.show()

Upsample the signal by an integer rate¶

When upsampling by 2, notice there are two output samples for every input sample.

resample_signal(2)

Input signal length: 100
Output signal length: 198

When upsampling by 4, notice there are four output samples for every input sample.

resample_signal(4)

Input signal length: 100
Output signal length: 396

Downsample the signal by an integer rate¶

When downsampling by 2, notice every other sample of the input appears at the output.

resample_signal(1 / 2)

Input signal length: 100
Output signal length: 50

When downsampling by 4, notice every fourth sample of the input appears at the output.

resample_signal(1 / 4)

Input signal length: 100
Output signal length: 25

Upsample by an irrational rate¶

When upsampling by $\pi$, notice there are roughly three output samples for every input sample. However, these samples often do not align with the original input samples.

resample_signal(np.pi)

Input signal length: 100
Output signal length: 312

Downsample by an irrational rate¶

When downsampling by $\pi$, notice there are roughly three input samples for every output sample. However, these samples often do not align with the original input samples.

resample_signal(1 / np.pi)

Input signal length: 100
Output signal length: 32