Radar as a Dot Product

In this article, we explore how radar systems perform one of the most challenging tasks in signal processing: detecting extremely weak echoes buried deep in noise. What appears as sophisticated engineering is actually an elegant implementation of dot products (or more formally an inner product).
If you are new to Ivethium, the following are some recommended articles before you start.
👉 Why Signals are Vectors
👉 The L2 Norm and Inner Products
👉 AM Radio as a Dot Product

Radar detection confronts us with one of signal processing's most brutal realities: finding signals that are so weak they're completely drowning in noise. Unlike AM radio, where we tune into relatively strong, continuous signals, radar must detect echoes that might be many times weaker than the background noise.

The mathematical beauty emerges when we realize that this seemingly impossible task is solved through a simple concept we've already explored: the inner product (also known as the dot product).

The Radar Transmission

Before we can detect anything, we need to understand what we're looking for. A radar system transmits a carefully designed pulse:

$$s(t) = w(t) \cdot \cos(2\pi f_c t)$$

Where:

• $w(t)$ is our pulse shape (the "pattern" we'll search for later)
• $f_c$ is the carrier frequency
• The pulse duration might be microseconds to milliseconds

This transmitted pulse $s(t)$ becomes our reference pattern. When it reflects off a target and returns to us, we'll be looking for this exact same pattern buried in noise.

What the Receiver Actually Sees

When the radar pulse reflects off a distant target and returns, the received signal looks like this (after coherent demodulation, similar to how we do it in a radio):

$$r(t) = A \cdot s(t - \tau) + n(t)$$

Where:

• $A$ is a tiny amplitude (the echo strength)
• $\tau$ is the time delay (tells us the target's range: $R = c\tau/2$)
• $n(t)$ is noise

Here's the challenge: $A$ might be 0.01 or smaller, while the noise $n(t)$ has amplitude around 1.0. The target echo is completely invisible.

But what exactly is this "noise" that's drowning out our signal? It's not just one thing - it's a chaotic mixture of:

• Thermal noise from the electronics themselves (atoms randomly vibrating in the receiver circuits)
• Atmospheric noise from lightning strikes around the world
• Cosmic radiation from space
• Ground clutter - reflections from buildings, trees, terrain
• Weather echoes - rain, snow, atmospheric phenomena
• Electronic interference - other radars, cell towers, etc.
• Our own radar's echoes - returns from previous pulses, reflections from other targets, multipath echoes bouncing off mountains or buildings

All of these sources combine into what appears to be random fluctuations with amplitude much larger than our tiny target echo. The received signal looks like pure chaos, with our precious target information buried somewhere inside this electromagnetic mess that includes our own radar's "pollution" too.

The Impossibility of Visual Detection

If you plotted the received signal $r(t)$ on an oscilloscope, you would see what appears to be random noise. The target echo, the thing we desperately need to find is buried so deep that no amount of staring at the waveform would reveal its presence.

This is fundamentally different from AM radio, where the modulated signal is clearly visible above the noise floor. In radar, the signal we seek is below the noise floor.

The Ambiguity Function: Our Mathematical Microscope

The solution lies in a mathematical tool called the ambiguity function, which is essentially a systematic way of computing inner products between our received signal and shifted versions of our transmitted reference.

For now, let's ignore Doppler and focus purely on time delays. The ambiguity function is:

$$\chi(\tau) = \int r(t) \cdot s^*(t - \tau) dt$$

Did you notice this equation is all about computing the inner product? We compute the dot product between:

• The received signal $r(t)$
• A time-shifted version of our transmitted signal $s^*(t - \tau)$

What this means physically: We're asking "How similar is the received signal to what we would expect from a target at range $R = c\tau/2$?" The factor of 2 comes from the round-trip nature of radar - the signal must travel to the target and back, so the total distance is $2R$ and the time delay is $τ = 2R/c$.

The Magic of Inner Product

Let's see what happens when we compute this dot product. Substituting our expression for $r(t)$:

$$\chi(\tau) = \int [A s(t - \tau_0) + n(t)] s^*(t - \tau) dt$$

$$\chi(\tau) = A \int s(t - \tau_0) s^*(t - \tau) dt + \int n(t) s^*(t- \tau) dt$$

The first integral is the inner product (informally we referred to this as "dot product") of our signal with itself (when $\tau = \tau_0$) - and we know from linear algebra that a vector's inner product with itself gives the maximum possible value.

The second integral is the inner product between random noise and our signal; since noise is random and our signal is structured, this averages toward zero.

The Inner Product Perspective

From our vector space viewpoint:

• The received signal $r(t)$ is a vector in signal space
• Each possible time-shifted reference $s(t - τ)$ is literally a different vector in that space
• The inner product $χ(τ)$ measures the inner product between the received signal vector and each reference vector

Even though the noise is much stronger than the signal, noise appears orthogonal to our desired signal in the signal vector space (Maybe I must write another article to explore why we can consider "noise" as an orthogonal component to our desired signal). Our target echo, however, points in one specific direction.

Please note that even perfect echoes that retain the exact waveform shape are different vectors compared to the transmitted pulse in Hilbert space since they are time-delayed.
Unless $τ=0$, the inner product $⟨s(t−τ), s(t)⟩$ will be less than the signal energy $\|s\|^2$, meaning there's an angle between the two vectors.

When we compute the inner product with the correct reference (when $τ = τ_₀$), we get the maximum possible value because we're computing the signal's inner product with itself. When we compute with incorrect references ($τ ≠ τ_₀$), the inner product is much smaller, probably closer to zero.

Signal Gain Through Integration

Here's the mathematical magic: when we compute the inner product over pulse duration $T$:

• Signal contribution: When $τ = τ_₀$, we get the maximum dot product value (signal with itself)
• Noise contribution: Random components average toward zero over the integration time
• Signal-to-noise improvement: Proportional to $√T$

This is called processing gain - the longer our integration time $T$, the more the random noise averages out while the signal remains at maximum strength.

The Pattern Recognition Challenge

What makes radar detection extraordinarily difficult is that we're performing pattern recognition under the worst possible conditions:

1. The Pattern Is Invisible

Unlike visual pattern recognition where we can see the object we're looking for, radar patterns are completely hidden. The received signal appears to be pure noise.

2. We Don't Know Where to Look

The target could be at any range from meters to hundreds of kilometers. This means we need to compute dot products for thousands of possible time delays:

$$χ(τ₁), χ(τ₂), χ(τ₃), ..., χ(τₙ)$$

Each computation is asking: "Is there a target at this specific range?"

3. False Alarms vs. Missed Detections

• Set the threshold too low → noise peaks trigger false alarms
• Set the threshold too high → real targets get missed

The mathematics of detection theory tells us this trade-off is fundamental and unavoidable.

Without FFT optimization, this would require computing each inner product separately - an enormous computational burden that would be impossible to achieve in real-time.

Discrete Implementation

The continuous ambiguity function becomes a discrete correlation:

$$χ[k] = Σ r[n] s^*[n - k]$$

Where:

• $k$ indexes the possible target ranges
• $n$ indexes the time samples
• Each value of $χ[k]$ is the result of a dot product

Why We Use FFTs

While the mathematical theory is elegant, the computational reality is daunting. As an example, a radar system might need to:

• Search 1000 range bins (different values of $\tau$)
• Update 1000 times per second
• Process multiple receive channels

Computing the ambiguity function directly would require millions of dot product operations per second — each involving thousands of multiply-and-add operations.

The breakthrough comes from recognizing that we can compute all these inner products simultaneously using the Fast Fourier Transform (FFT).

Note: We do not consider FFT as a part of the core Ivethium-style deep explorations, because the existing material on FFTs is already well-established and intuitive. However, the idea that time–frequency duality enables computational shortcuts is important, especially when leveraging structural symmetry in physical systems. The FFT is a manifestation of that principle.

The ambiguity function computation:

$$\chi(\tau) = \int r(t) s^*(t - \tau) dt$$

is equivalent to:

$$\chi(\tau) = \text{IFFT} \left[ \text{FFT}(r) \text{conj} \left( \text{FFT}(s) \right) \right]$$

This transforms the problem into the frequency domain, where:

• We decompose both signals into their frequency components
• We multiply corresponding frequency components (element-wise)
• We transform back to get all correlation values at once

The computational savings are enormous: instead of $N^2$ operations for $N$ range bins, we need only $N \log N$ operations — a speedup of potentially 100× or more for typical radar systems.

Adding Doppler: The Two-Dimensional Search

Now we can introduce the complication we initially avoided. If the target is moving, the returned echo will have a frequency shift (Doppler effect). A frequency shift in the time domain is represented by multiplying by a complex exponential.

🔍 Why Does a Frequency Shift Correspond to Multiplication by a Complex Exponential?

This is a fundamental property of the Fourier Transform. If a signal $s(t)$ is shifted in frequency by $f_0$, the shifted signal becomes:

$$s_{\text{shifted}}(t) = s(t) \cdot e^{j 2\pi f_0 t}$$

Why does this work? Because the Fourier transform decomposes a signal into sinusoids. It sees the world in frequency. When you multiply by $e^{j 2\pi f_0 t}$ in the time domain, you're rotating every frequency component by $f_0$ — effectively shifting the entire spectrum.

This is a direct result of the modulation property of the Fourier Transform:

$$\mathcal{F}(s(t) e^{j 2\pi f_0 t}) = S(f - f_0)$$

In essence, time-domain multiplication becomes frequency-domain translation. This duality how multiplication in one domain becomes shifting in the other is at the heart of why modulation, demodulation, and Doppler shifts all operate via simple complex exponentials.

The received signal becomes:

When a target is moving, its return signal experiences a Doppler frequency shift. This is modeled as:

$$r(t) = A \cdot s(t - \tau) \cdot e^{j 2\pi f_d t} + n(t)$$

Here:

• $A$ is the amplitude (possibly complex)
• $\tau$ is the time delay (related to target range)
• $f_d$ is the Doppler frequency shift
• $n(t)$ is additive noise
• The complex exponential $e^{j 2\pi f_d t}$ represents a frequency shift applied to our time-delayed signal.

Now, the ambiguity function becomes two-dimensional:

$$\chi(\tau, f_d) = \int r(t) s^*(t - \tau) \cdot e^{-j 2\pi f_d t} dt$$

This corresponds to computing inner products across a 2D grid:

• Horizontal axis: possible ranges (time delays $\tau$)
• Vertical axis: possible velocities (Doppler frequencies $f_d$)

Each point $(\tau, f_d)$ represents a specific hypothesis:

“There is a target at range $R = \frac{c \tau}{2}$ moving with velocity $v = \frac{c f_d}{2 f_c}$.”

The value

$$|\chi(\tau, f_d)|^2$$

indicates how strongly the data supports that hypothesis. This creates a delay–Doppler map where brighter regions highlight likely targets.

Conclusion: The Mathematics of Finding the Unfindable

Radar detection demonstrates the power of mathematical frameworks to solve problems that seem impossible from an intuitive perspective. While a signal buried below the noise floor appears completely undetectable, the mathematics of inner products provides a systematic way to extract it. The key insights that make this possible are:

1. Structured vs. Random: Our signal has structure; noise is random
2. Coherent Integration: Signal adds coherently; noise adds randomly
3. Optimal Correlation: The dot product is mathematically optimal for this detection problem
4. Processing Gain: Integration time provides √T improvement in detectability

Every radar detection is fundamentally an inner product computation, asking: "How similar is this received data to what I would expect from a target at this location?"

The ambiguity function, despite its mathematical complexity, serves one simple purpose: it systematically computes these inner products across all possible target locations, creating a map where targets appear as peaks rising above the noise floor.

In radar, the difference between a successful inner product and a failed one can determine whether an aircraft is detected, whether an autonomous vehicle avoids a collision, or whether a weather system is tracked accurately. The mathematics of inner products, pushed to their absolute limits, become the foundation for technologies that quite literally save lives by finding the unfindable.