sdr.frequency_offset(x: ArrayLike, offset: ArrayLike, offset_rate: float = 0.0, phase: float = 0.0, sample_rate: float = 1.0) → NDArray

Applies a frequency and phase offset to the time-domain signal \(x[n]\).

Parameters:
x: ArrayLike

The time-domain signal \(x[n]\) to which the frequency offset is applied.

offset: ArrayLike

The frequency offset \(\Delta f = f_{\text{new}} - f\) in Hz.

offset_rate: float = 0.0

The frequency offset rate \(\Delta^2 f / \Delta t\) in Hz/s. For example, a frequency offset varying from 1 kHz to 2 kHz over 1 ms, the offset rate is 1 kHz / 1 ms or 1 MHz/s.

phase: float = 0.0

The phase offset \(\phi\) in degrees.

sample_rate: float = 1.0

The sample rate \(f_s\) in samples/s.

Returns:

The signal \(x[n]\) with frequency offset applied.

Notes

The frequency offset is applied by multiplying the signal by a local oscillator. The local oscillator is calculated as follows.

\[\begin{split} f = \Delta f[n] + \frac{\Delta^2 f[n]}{\Delta t} t \\ \text{lo}[n] = \exp \left( j \left[ 2 \pi f t + \phi \right] \right) \\ y[n] = x[n] \cdot \text{lo}[n] \end{split}\]

Examples

Create a reference signal with a constant frequency of 1 cycle per 100 samples.

In [1]: x = sdr.sinusoid(100, freq=1 / 100)

Add a frequency offset of 1 cycle per 100 samples (the length of the signal). Notice that the signal now rotates through 2 cycles instead of 1.

In [2]: freq = 1 / 100

In [3]: y = sdr.frequency_offset(x, freq)

In [4]: plt.figure(); \
   ...: sdr.plot.time_domain(np.unwrap(np.angle(x)) / (2 * np.pi), label="$x[n]$"); \
   ...: sdr.plot.time_domain(np.unwrap(np.angle(y)) / (2 * np.pi), label="$y[n]$"); \
   ...: plt.ylabel("Absolute phase (cycles)"); \
   ...: plt.title("Constant frequency offset (linear phase)");
   ...:
../../_images/sdr_frequency_offset_1.svg

Add a frequency rate of change of 2 cycles per 100^2 samples. Notice that the signal now rotates through 4 cycles instead of 2.

In [5]: freq_rate = 2 / 100**2

In [6]: y = sdr.frequency_offset(x, freq, freq_rate)

In [7]: plt.figure(); \
   ...: sdr.plot.time_domain(np.unwrap(np.angle(x)) / (2 * np.pi), label="$x[n]$"); \
   ...: sdr.plot.time_domain(np.unwrap(np.angle(y)) / (2 * np.pi), label="$y[n]$"); \
   ...: plt.ylabel("Absolute phase (cycles)"); \
   ...: plt.title("Linear frequency offset (quadratic phase)");
   ...:
../../_images/sdr_frequency_offset_2.svg