CSI, one picture at a time.

Step 1 of 12

A wave is a repeating number

An antenna carries a changing electrical current. It produces an electromagnetic field; a receiving antenna converts part of that field back into a voltage. A sine wave is the simplest description of one repeating component.

A graph needs a horizontal variable and a vertical quantity. Here the horizontal axis is time and the vertical axis is a normalized voltage. The height is amplitude A. Frequency f counts cycles per second (hertz). Phase φ tells us where in the cycle we start. One turn is 360° or 2π radians; π radians is half a turn.

A real radio oscillates billions of times per second. This figure is deliberately slowed to a few cycles so you can see it. A time plot is not a picture of a wiggly trajectory through the room. In space the wavelength is λ = c/f; c = 299,792,458 m/s in vacuum. Indoor propagation also involves materials, reflections and antenna patterns.

v(t) = A cos(2π f t + φ)     T = 1/f     λ = c/f

Try it in the lab. Change amplitude, then phase, then frequency. Which control changes the number of cycles, and which only moves the peaks?

Step 2 of 12

Two coordinates describe one wave

A complex number is a pair of real numbers with useful arithmetic. Draw I horizontally and Q vertically. The arrow from zero to (I,Q) has a length and an angle.

Write H = I + jQ, with j² = −1. The word imaginary names the vertical coordinate; it does not mean the measurement is imaginary. Pythagoras gives the length √(I²+Q²). atan2(Q,I) finds the angle while keeping the correct quadrant. At (0,0), the angle is undefined.

Euler's identity, e^(jφ) = cos φ + j sin φ, gives the same number in polar form A e^(jφ). Its real rotating projection is I cos(2πft) − Q sin(2πft). I and Q are two coefficients for one waveform, not two antennas, packets or people. The 3D helix adds time as a third mathematical axis; it is not the spatial path of the radio wave.

H = I + jQ = |H| e^(jφ)     |H| = √(I²+Q²)     φ = atan2(Q,I)

Try it in the lab. Set I = 3 and Q = 4. Read the 5-unit arrow. Switch to 3D, rotate it, and follow the same complex number through time.

Step 3 of 12

A channel scales and rotates

Send a known complex symbol X through a channel H. In a simple, narrowband, noiseless model, the received symbol is Y = HX. Complex multiplication multiplies lengths and adds angles.

H is the channel's effect, X is what was sent, and Y is what arrived. They are different quantities even though all three can be complex numbers. A CSI coefficient is an estimate of H, not a payload bit, an instantaneous antenna voltage, or a location.

One coefficient works for a frequency narrow enough that the channel is approximately constant over it. Wi-Fi uses many such frequencies. Real reception also has noise and interference: Y = HX + N. If H is nearly zero, dividing by it strongly amplifies the noise. A deep fade is therefore difficult to equalize.

Y = H X + N     |HX| = |H| |X|     arg(HX) = arg(H) + arg(X)

Try it in the lab. Rotate H by 90° and reduce its amplitude. Watch X stay fixed while Y moves. Then make the gain approach zero.

Step 4 of 12

Known training symbols reveal H

The receiver cannot infer both an unknown message and an unknown channel from one received value. Wi-Fi packets carry known training fields. The receiver compares what should have arrived with what did arrive.

For a nonzero known training symbol Xₖ, the simplest least-squares estimate is Ĥₖ = Yₖ/Xₖ. Repeated training can be combined to reduce noise. In MIMO, training across streams allows a matrix estimate; it is not independent division of an arbitrary mixture.

The packet path is: antenna → RF filtering and amplification → downconversion into I/Q → sampling → timing/frequency synchronization → FFT → training-field channel estimate → equalization and decoding. CSI extraction exposes an internal estimate. The training field, packet PHY, firmware and capture configuration determine what is exported. Later pilots can track residual phase errors; they are not extra antennas.

Ĥ = (Σ X*Y)/(Σ |X|²)     for one X: Ĥ = Y/X

Try it in the lab. Increase the deterministic noise amplitude and then the number of known training symbols. Compare the estimate with the planted channel.

Step 5 of 12

A vector samples the channel across frequency

A Wi-Fi channel contains many closely spaced subcarriers. They add into one transmitted waveform. After an FFT, the receiver estimates a complex gain at each reported frequency.

An index i is a position in an array. A subcarrier number k is a signed FFT-bin coordinate relative to the RF center f꜀. The physical frequency is fₖ = f꜀ + kΔf. Negative k means below the center, not negative radio frequency. The center (DC) and guard tones are not ordinary transmitted data tones. Pilots are known symbols on active tones.

With useful OFDM symbol duration Tᵤ, Δf = 1/Tᵤ. Legacy/HT/VHT use 312.5 kHz; HE uses 78.125 kHz. HE has denser sampling at the same bandwidth. That does not by itself improve delay resolution: bandwidth, not the number of stored coefficients alone, sets that scale. Arrays may omit nulls, include them, group tones, or contain several training fields. Always read metadata.

H = [H[k₀], H[k₁], …]     fₖ = f꜀ + kΔf     Δf = 1/Tᵤ

Try it in the lab. Switch between 52 used legacy tones, 242 used HE20 tones, and the C5's 53-entry L-LTF. Select an array position and locate its real frequency.

Step 6 of 12

Every coefficient adds many physical paths

The receiver hears a direct path plus reflected, scattered and diffracted copies. A longer path arrives later. At a given frequency, those copies add as complex arrows.

A path with delay τ contributes a phase −2πfτ and a complex amplitude a. Sum all paths to get H(f). A tiny movement can alter their relative phases; two strong paths may cancel. A lower amplitude therefore does not uniquely mean a person blocked the direct line. A person can change both amplitudes and delays, and can affect links without standing on their straight line.

The lab uses one direct ray and one floor-reflected ray in an authored room, with a chosen echo gain. Heights and lengths are in metres. The mirror construction gives the specular reflection point. It omits diffraction, wall materials, antenna patterns and multiple bounces. The direct path's common delay is removed, leaving relative phase. Drag the 3D view or switch to plan view; moving the receiver changes the same model in both.

H(f) = Σ aₗ exp(−j2πfτₗ)     τₗ = path lengthₗ/c

Try it in the lab. Move the receiver height by a few centimetres. Watch the path difference and the selected tone change. Then set echo gain to zero.

Step 7 of 12

Antennas add a spatial dimension

Each receive antenna samples the field at a different position and polarization. Each distinguishable transmit stream can have a different effective channel to it. At one frequency, collect these links into a matrix.

In an ideal physical-antenna basis, Yᵣ = Σₛ HᵣₛXₛ + Nᵣ. A 2×2 channel has four complex coefficients per tone. Stack K tones and P packets to obtain H[packet, receive, transmit, tone]. Software can transpose or flatten this tensor without changing the physics. The project's analysis view is [record, tone, chain], where chain = rx·ntx + tx.

A transmitter may map spatial streams through precoding or cyclic shifts. Exported CSI can then be an effective stream channel rather than a separately calibrated physical antenna channel. The AX210 has two receive/transmit chains, but a one-stream packet does not automatically yield a measured 2×2 matrix. Use nrx, ntx and validity from the record. Switched antennas on a single RF chain are not simultaneous MIMO.

Mount antennas rigidly; document cable/connector, spacing, orientation, polarization, height and band. Half-wavelength spacing is an array-design scale, not a guarantee of angle estimation. Phase differences also contain cable and hardware offsets and multipath. Moving an antenna or touching its cable changes the instrument. Two disconnected receivers do not automatically share a phase reference.

Y[k] = H[k] X[k] + N[k]     chain = rx·Ntx + tx

Try it in the lab. Select a receive antenna, a transmit stream and a tone. Follow the highlighted matrix cell and the flattened chain index.

Step 8 of 12

Read every number in a vendor vector

The physics is shared; storage conventions are not. Choose an explicit profile, then inspect one coefficient from a synthetic payload or paste your own signed scalar words. Everything stays in this browser.

The AX210/iax CSIQ profile stores contiguous chain blocks. Each coefficient is a little-endian signed 16-bit imaginary word followed by a real word. Its payload byte offset is 4·(chain·K + i). File headers and TLV headers are outside this offset. The C5 L-LTF SDK profile has 53 signed 8-bit pairs ordered k = 0…26, −26…−1. The DC entry is not a channel measurement. Plotting can reorder it, but storage position and k must remain distinguishable.

Select the profile that actually matches the recording. C5 HT/VHT20 and HE20 SU grids can contain 57 and 245 entries including nulls; STBC, MU, bandwidth and LTF acquisition settings need their own interpretation. This lab intentionally accepts only one legacy C5 L-LTF block, or the stated AX210 used-tone blocks. It does not guess an unknown packet format from vector length.

Before using an AX210 chain, check the record's measured-chain mask: parsed RSSI −127 dBm marks a stale unmeasured block. C5 ESP-IDF v6.0 documents rx_channel_estimate_info_vld; consult the exact SDK, rather than borrowing first_word_invalid rules from another ESP32. This inspector has no packet header and cannot validate those flags. It explains numbers, not capture integrity. Raw amplitudes from different vendors have no common calibrated power scale.

H = real + j·imag     word offset = 2(chain·K+i)     byte offset = word offset·word size

Try it in the lab. Inspect the pair [4,3]. Then enable the deliberate I/Q mistake: its magnitude stays 5 but its phase changes. Paste a matching complete payload to inspect your own values.

Step 9 of 12

Measured phase includes the receiver

A radio's clock is not perfectly synchronized to the other radio. The reported phase includes propagation, a common phase rotation and a slope across frequency, as well as noise and hardware effects.

Phase wraps at ±π: a jump from +179° to −179° is a 2° change. Unwrapping chooses a continuous sequence by adding whole turns; it cannot recover arbitrary delays or fix missing data. A simplified impairment model adds α per packet and βk across subcarriers. CFO contributes evolving common phase; sufficiently large CFO also creates intercarrier interference, beyond this simple model. Timing and sampling offsets contribute slope.

Sanitizing by subtracting a fitted line removes the common offset and slope, but also removes genuinely linear channel phase. A clean residual is not absolute propagation delay. A same-packet ratio Hₐ/Hᵦ or conjugate product HₐHᵦ* can cancel shared phase errors. Chain-specific errors remain; a ratio becomes unstable when its denominator is near zero. Separate ESP receivers do not share those errors just because they hear the same AP.

φmeasured(k,p) ≈ φchannel(k,p) + α(p) + β(p)k     arg(HₐHᵦ*) = φₐ − φᵦ

Try it in the lab. Add common rotation and timing slope. Compare wrapped, unwrapped and detrended views. Notice that removing slope also erases the planted physical delay.

Step 10 of 12

A waterfall is a stack of packet spectra

One packet supplies one frequency response. Repeat the measurement and stack the rows. A waterfall turns amplitude into color so you can see which frequencies change at which times.

The horizontal axis is subcarrier frequency offset in MHz; the vertical axis is elapsed time in seconds. Each colored cell represents one time and one frequency. The color is 20 log₁₀|H| relative to a stated amplitude reference, not dBm. A 3D surface shows the same values with height as an additional encoding. It is not a 3D scan of the room.

Here one echo rotates slowly while a static path remains. This makes time-varying interference even though the transmitter's data symbols are not plotted. Set motion frequency to zero for a static channel. Real rows arrive at packet times, not at a guaranteed rate. Scope by record class and transmitter before interpreting a stripe; changing packet types, transmitters, gain settings or missing records can create apparent motion.

A[p,k] = 20 log₁₀(|H[p,k]| / reference amplitude)

Try it in the lab. Switch 2D to 3D and select a packet. The orange spectrum below must match that exact row. Freeze motion at 0 Hz.

Step 11 of 12

Frequency structure becomes a delay profile

The inverse Fourier transform combines frequencies into a delay-domain description. A delayed path produces a phase slope across frequency; combining those slopes creates a peak near its delay.

The ideal channel impulse response h(τ) is a sum of delayed path contributions. The finite measured band blurs each path into a lobe. The lab uses an ideal contiguous frequency grid to isolate this idea. It applies a Hann window and computes a zero-padded inverse transform. Increasing zero padding makes the curve smoother without resolving closer physical paths.

The delay resolution scale is about 1/B for observed bandwidth B; windowing widens the main lobe. At 20 MHz, 1/B = 50 ns, about 15 m of path length cτ, not 15 m of guaranteed location accuracy. Active-tone bandwidth is smaller than nominal bandwidth. csiscope uses a Hann-windowed IFFT, and its legacy contiguous-tone approximation and packet timing offsets preclude interpreting an absolute peak as calibrated range. An uncalibrated peak shift can be receiver timing.

h[n] = IFFT{W[k]H[k]}     PDP[n] = |h[n]|²     delay scale ≈ 1/B

Try it in the lab. Try echoes 20 ns apart, then 150 ns apart. Increase bandwidth, then padding. Only bandwidth changes the underlying ability to separate delays.

Step 12 of 12

Doppler is change across packets

Subcarrier frequency describes where we listen within the radio channel. Doppler frequency describes how a channel coefficient evolves over time. These are two different Fourier axes.

For one path with changing length L(t), phase is −2πL/λ. The signed Doppler shift is fᴅ = −(1/λ)dL/dt under this convention. Bistatic motion changes transmitter-to-object plus object-to-receiver length. The sign and magnitude depend on geometry; speed is not universally λfᴅ/2. Tangential movement can have little instantaneous Doppler.

Sample at fₛ packets per second and an ideal uniformly sampled complex signal is ambiguous outside ±fₛ/2. The lab plots the true slow-time sinusoid and the indistinguishable aliased reconstruction through its samples. csiscope's spectrogram uses an STFT of a two-chain conjugate product. Its hertz axis also depends on real arrival times, missing slots and resampling. Check the metronome and arrivals before calling a ridge motion; a packet cadence can create a ridge too.

fᴅ = −(1/λ)dL/dt     unaliased interval: −fₛ/2 ≤ fᴅ < fₛ/2

Try it in the lab. Use 12 Hz channel change at 20 packets/s: it aliases to −8 Hz. Raise the delivered sampling rate. Changing the RF carrier is a different operation.