Radar Echo Demo: How Convolution and Correlation Emerge

Transmit a pulse, measure the environment response, reconstruct echoes by convolution, and then compare sent and received signals by cross-correlation to estimate alignment and lag.

Radar scene

Move the target farther away and the returned echo arrives later. The sequence length is fixed at 100 so you can clearly see the lag. We treat the environment as a linear system: if we know its response h[n] to a unit impulse, then by superposition we can estimate its response to any transmitted signal.
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Transmitted signal

Start with a single delayed pulse, then try richer transmitted waveforms. Any discrete transmitted sequence can be written as a train of scaled and shifted unit pulses.
Convolution goal: if we know the response h[n] to a unit impulse and assume linearity, then for u[n] = Σₖ u[k]δ[n-k] we estimate the echo by y[n] ≈ Σₖ u[k]h[n-k]. Correlation goal: measure how strongly the sent and received signals match at each lag, for example rxy[ℓ] = Σₙ x[n]u[n-ℓ].
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Direct echo x[n]
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Convolution y[n]
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1. Virtual radar scene

Radar transmitter/receiver
Current transmitted sample
Current echo sample
Target / reflector

2. Measured echo signature h[n]

First click Measure echo signature h[n]. This sends a unit pulse δ[n] and records the returned echo. The central idea of convolution is then: if the system is linear, this measured unit response is enough to build the response to any input by superposition.

3. Transmitted signal u[n]

Any discrete transmitted signal can be written as a weighted impulse train: u[n] = Σₖ u[k]δ[n-k]. Each transmitted pulse generates its own delayed and scaled copy of the measured echo signature.

4. Received echo: direct simulation vs convolution

Blue is the directly simulated received echo. Purple is the response reconstructed from h[n] by superposition. Convolution answers this question: given the unit impulse response, what would the system return for an arbitrary transmitted signal?

5. Cross-correlation: matching the sent and returned pulses

Cross-correlation measures how strongly the sent signal and the received signal match as one is shifted relative to the other. In dot-product form, for lag ℓ, rxy[ℓ] = Σₙ x[n]u[n-ℓ]. When the shifted transmitted pulse lines up with the returned echo, the dot product becomes large. Convolution uses h[n-k]. Correlation uses u[n-ℓ]. The sign flip appears because convolution builds an output from shifted impulse responses, while correlation slides one signal against another to measure alignment.
Conceptual explanation

Imagine there is a radar and a target somewhere in the environment. The radar sends out a short pulse of energy. After travelling through the environment, part of that energy reflects from the target and returns or “filters back” to the radar as an echo. The time it takes for the echo to return determines the delay, which is related to the distance of the target.

In the demo, the radar first sends a unit pulse u[n] = δ[n]. The environment responds with a returned signal h[n]. This measured signal is called the impulse response of the environment. It tells us how the environment reacts to a single transmitted pulse.

If the radar sends the same pulse later in time, the echo is simply a delayed copy of the same response. If the transmitted pulse is scaled, the echo scales as well. These observations reflect the assumptions of a linear time-invariant system.

Any transmitted discrete signal can be written as a sum of scaled and shifted impulses
u[n] = Σₖ u[k] δ[n-k]

Each of these impulses generates its own delayed copy of the impulse response. Because we assume the system is linear, the response to a sum of inputs is equal to the sum of the responses to each input separately. As a result, the total echo returned by the environment is simply the sum of all these delayed impulse responses.
y[n] ≈ Σₖ u[k] h[n-k]

This operation is called convolution. Convolution answers the question (for a linear system): If we know how the environment responds to or “filters” a single pulse, what signal will we receive for any transmitted waveform?

The final panel illustrates cross-correlation. Instead of predicting the returned signal, correlation measures how strongly the transmitted signal and the received signal match when one is shifted relative to the other. In dot-product form
rxu[ℓ] = Σₙ x[n] u[n-ℓ]

When the shifted transmitted pulse aligns with the returned echo, the dot product becomes large. The location of the peak therefore reveals the delay between transmission and echo, which corresponds to the distance of the target.

In summary, convolution models how a system transforms or filters a signal using its impulse response, while correlation measures how well two signals align as one is shifted relative to the other.

These two operations are closely related. If we flip one of the signals in time, then correlation becomes convolution:
(x ⋆ u)[ℓ] = (x * u[-n])[ℓ]

So convolution and correlation are not unrelated ideas. They differ mainly in purpose and interpretation: convolution builds the output of a system from its impulse response, while correlation measures similarity by sliding one signal against another. The time reversal explains the sign flip between the two formulas.
(c) Fayyaz Minhas