FIR Filter & LTI System Lab
A visual workflow for understanding Discrete-Time Convolution
1 Define Input Signal $x[n]$
This is your raw data. In an LTI system, we want to modify this signal's characteristics.
2 Define Filter $h[k]$
These coefficients determine the Impulse Response. This defines how the system reacts.
3 The "Sliding Window" Mechanism
The filter output at index $n$ is the sum of products:
$$y[n] = \sum_{k=0}^{M-1} h[k] \cdot x[n-k]$$
4 Time Domain Analysis
5 Frequency Response $H(e^{j\omega})$
Why is this an LTI System?
1. Linearity
If you process Signal A and Signal B separately, then sum them, the result is the same as processing (A + B). Mathematically: $\mathcal{H}\{ax_1 + bx_2\} = a\mathcal{H}\{x_1\} + b\mathcal{H}\{x_2\}$.
2. Time Invariance
The filter coefficients $h[k]$ do not change over time. If you input a signal today or tomorrow, the filter's "rule" remains identical.