Matched Filter Online Simulator

Matched Filter: Step-by-Step

How do we find a "1" inside a mountain of noise? Let's break it down.

1. Data Input (Bits)

2. Channel Noise (SNR) 5 dB

3. Filter Shape Sine Wave (Standard) Square Pulse

STEP 1: The Template

To recognize a signal, the receiver needs a "Search Image" (The Template). We create one perfect cycle of our carrier wave.

Logic: If the incoming signal matches this shape, the math will result in a MAXIMUM PEAK.

STEP 2: The Noisy Reality

Here is your data sent over the air. We added Random Noise. Notice how the "1" and "0" shapes start to disappear.

Sent:

STEP 3: The Convolution (Sliding)

We slide our Template across the Noisy Signal. At every point, we multiply them and sum them up.

What's happening? When the template perfectly aligns with a '1', the product is positive $\times$ positive = Huge Positive.

STEP 4: Sampling & Decision

We only look at the output at the exact end of the bit period (The Vertical Lines). This is where the peak is highest.

Detected:

Decision Rule:
If Value $> 0 \rightarrow$ Bit 1
If Value $< 0 \rightarrow$ Bit 0

The "Deep" Math Secret

Why does the Matched Filter work better than just looking at the signal?

Because noise is random. When you sum (integrate) a signal over time:

• The Signal is consistent, so it adds up linearly: $S + S + S = 3S$.
• The Noise is random, so it partially cancels itself out: $+N - N + n - n \approx 0$.

This is called Processing Gain. The longer the bit lasts, the easier it is to see the peak!

Why does the Filter work?

1. The Signal is "Consistent"

Because the BPSK wave follows a pattern (the Sine wave), every part of the wave "agrees" with the filter.

Example (adding 4 steps):
(+10) + (+10) + (+10) + (+10) = +40

The result "adds up linearly" (it grows bigger and bigger in a straight line).

2. The Noise is "Random"

Noise has no pattern. At any moment, it might jump Up or Down. It is chaotic.

Example (adding 4 steps):
(+5) + (-6) + (+2) + (-1) = 0

The pluses and minuses "cancel each other out," staying near zero.

The Matched Filter Equation

To separate the signal from the noise, we use the Discrete Convolution formula:

$$y[n] = \sum_{k=0}^{L-1} x[n-k] \cdot h[k]$$

y[n]
The Output Peak

=

x[n-k]
Noisy Input

×

h[k]
The Template

By multiplying the input by a "Template" ($h[k]$), we ensure that only the parts that look like our signal get amplified, while the noise gets crushed.