Least Squares FIR Filter Design: Applications in DSP Wireless System Design

This repository contains Digital Signal Processing (DSP) algorithms implemented in Python. We begin with the implementation of a basic finite impulse response (FIR) filter using the Least-Squares (LS) estimation method to estimate the filter coefficients.

What is Least-Squares FIR Filter Design?

Least Squares (LS)-based estimation is a method of estimating the frequency response of a discrete-time (DT) system. **The total channel (Tx + Channel + Rx) response can be modeled using a finite impulse response (FIR) filter:

$$ y[n] = \sum\limits_{k = 0}^{L-1} h[k]. x[n-k] $$

In the above, $$x[n]$$, $$y[n]$$, and $$h[k]$$ represent known transmitted samples, captured received samples, and an FIR filter of length $$L$$, respectively.

Parameter Estimation

Step 2: Preparing the input Convolution Matrix

From the known input samples $$x[n]$$, we need to form the convolution matrix X:

• $$x$$ = $$[x[0], x[1], ..., x[N-1]]^T$$
• For an FIR filter of length $$L$$:

$$ \begin{pmatrix} x[L-1] & x{L-2} & \cdots & x[0] \\ x{L} & x[L-1] & \cdots & x[1] \\ \vdots & \vdots & \ddots & \vdots \\ x[N-1] & x[N-2] & \cdots & x[N-L] \\ \end{pmatrix} $$

In the above, matrix X represents the sliding windows over the input signal used in convolution.

Step 3: Forming the Input-Output Vector The received vector after digitization is aligned with the input vector to match each row of X. The output vector y can be given by: $$ \begin{bmatrix} y[L-1] \ y{L} \ \vdots \ y[N-1] \ \end{bmatrix} $$

Step 4: Least-Squares Solution

Importance in DSP and Wireless System Design

• Channel Estimation: estimating the channel impulse response of a multipath wireless channel and performing channel equalization at the receiver for phase, time, and carrier synchronization.
• Channel Equalization: estimating the inverse of the channel response to compensate for interference at the receiver, for example, compensating inter-symbol interference (ISI) in an orthogonal frequency division multiplexing (OFDM) receiver.
• Echo Cancellation: modeling transmitter leakage channel in full-duplex radios simultaneously transmitting and receiving (STAR) at the same frequency band, and cancelling self-interference (SI) signal at the digital baseband.