Modifying the Polyphase Filter Bank to make it robust to quantization effects

The polyphase filter bank (PFB) is a widely used digital signal processing tool used for channelizing input from radio telescopes. Quantization of the channelized signal leads to blow ups in error. We present a practical method for inverting the PFB with minimal quantization-induced error that requires as little as 3% extra bandwidth.

• pfb.py contains fuctions to perform the forward and inverse PFB, and methods to quantize the inverse.
• helper.py utility functions used to analyzing the quantization errors induced in the quantized iPFB.
• conjugate_gradient.py contains functions to optimize the chi-squared value of the iPFB.
• matrix_operations is a helper that wraps PFB related operations in a way that makes them look like the linear operators that they really are. These are used in conjugate gradient descent algorithm.
• optimal_wiener_thresh.py finds the optimal Wiener threshold parameter.
• plots/plotall.sh generates all plots (which can be found in plots/img).

• Jax, for autograd custom gradient descent functions

The usual suspects

• Numpy
• Scipy (scipy.signal, scipy.optimize)
• Matplotlib

Let $X$ be a gaussian random variable with $\mu=0$ and $\sigma=1$, and let $(x_n)$ be a sequence of i.i.d realizations of $X$. If we would like to quantize this signal to four bits, what is the optimal quantization interval? This is not a mathematically precise question because it depends on what your optimizing. Lets say we wish to minimize the expected magnitude squared of the residual $R=X-\tilde X$, where $\tilde X$ is a quantized signal.

Let $(y_n)$ be the normalized FFT of $(x_n)$ defined as follows

$$y_n = \frac{1}{\sqrt N} \sum_{k=0}^N \exp(-2\pi i nk)x_k$$

In conjugate_gradient.py, we have code that lets us optimize the inverse PFB based on some added information.

We perform conjugate gradient descent on a matrix equation of the form

$$B x = u$$

The equation we are minimizing, the chi-squared equation, takes the form

$$\chi^2 = (d - Ax)^T N^{-1} (d - Ax)$$

Taking a derivative wrt to the model ($x$) and setting that to zero we get

$$\frac{d\chi^2}{dx} = -2A^TN^{-1}(d - Ax) = 0 \Rightarrow A^TN^{-1}(d - Ax) = 0$$