A sonar records a real voltage on a carrier, and its processor works on complex samples near zero frequency. Nine steps follow one echo from the hydrophone to complex baseband and back, and end at the carrier phase that back-projection has to restore.
The step opens with the rotation off. Choose Correct to watch the phasors line up and the seabed focus.
I · real · +f Q · imaginary · −f result reference overlap
Time
Spectrum
Quadrature demodulator
I/Q plane
Real numbers per second of signal
Toy image, 20 cm along-track
Along-track up, slant range to the right, 10 cm across; dB against the correctly rotated peak. Click a pixel or use the arrow keys to choose the one whose phasors are drawn.
Baseband data, ping against range
Brightness is magnitude and color is phase (key at right); one row per ping, along-track up, and 30 cm of range to the right.
ApertureLab image of the patch
Point target alone, 1.2 m
Toy phasor chain at the chosen pixel
Overview
Readers come to this subject from two directions. In acoustics the data are a real voltage on a carrier, which is what the hydrophone produces, and converting them to complex numbers near zero frequency can look like an unnecessary transformation. In signal processing the complex samples are the starting point, and it is easy to lose sight of the fact that their phase measures a path length; that fact is why a back-projection beamformer must multiply every sample by the carrier phase of its delay.
The steps above connect the two descriptions with one echo. The sonar is the one the rest of this site describes: a 300 kHz carrier, 60 kHz of bandwidth, a compressed pulse with a Hann-weighted spectrum, and a point on a flat bed 16 m out in ground range from a track 10 m up. The ratio of carrier to bandwidth is 5, and it recurs throughout: five carrier cycles per resolution cell in delay, and five turns of phase per resolution cell in range.
Step 1
The transducer can only radiate efficiently in a band around its resonance, so the transmitted pulse is a burst of carrier, and the echo is a burst of carrier delayed by the two-way travel time. After pulse compression its envelope is about 1.44/B wide at −3 dB, which holds about seven carrier cycles.
The echo carries two kinds of information. The envelope says when the echo arrived, to within a resolution cell of c/2B = 12.5 mm of range. The carrier's phase under the envelope says where within that cell, to a small fraction of a wavelength. Moving the target 1.25 mm moves the envelope by a tenth of a cell and slides the carrier by half a cycle.
Step 2
A phasor ejωt is a point turning counterclockwise on the unit circle; its mirror e−jωt turns clockwise at the same rate. Their sum stays on the real axis, because the imaginary parts cancel at every instant:
The clockwise phasor is what a negative frequency means. Every real signal is built from such pairs, so its spectrum at −f is the complex conjugate of its spectrum at +f, and the half of the spectrum below zero frequency repeats the half above it. In the helix the two phasors trace a right-handed and a left-handed corkscrew whose sum lies flat on the real wall.
Step 3
Sampling at a rate fs repeats the spectrum every fs. To keep the copies apart by sampling above twice the highest frequency, the echo needs more than 660 k samples per second, although its band occupies 270 to 330 kHz and the spectrum between zero and 270 kHz is empty. The cost is set by the carrier, and the information is set by the bandwidth.
A real signal can be sampled below its carrier if the rate is chosen so that the copies interleave without overlapping. For this band the valid rates form windows, the narrowest from 132 to 135 kHz, where the band folds down to near zero intact. Outside the windows the copies overlap, and nothing recovers the signal afterwards.
Step 4
Multiplying two tones gives their sum and difference frequencies: cos a·cos b = ½cos(a−b) + ½cos(a+b). The same identity read in the other direction explains beats: two tuning forks sounding together add, and their sum equals a tone at the mean frequency times a slow cosine, so the loudness swells at the difference frequency although no tone at that frequency is present. Multiplying the echo by a cosine at f0 therefore moves every spectral component both up and down by f0: the band at +300 kHz lands at 0 and at +600 kHz, and its mirror at −300 kHz lands at 0 and at −600 kHz.
The two copies that land at zero overlap, and they are mirror images of each other. The spectrum panel uses a test signal with a strong tone 10 kHz above the carrier and a weaker one 20 kHz below it, so that the two sides can be told apart: after a cosine mixer each tone appears at both plus and minus its offset, and whether it was above or below the carrier can no longer be read from the output.
Step 5
Multiplying by e−j2πf0t shifts every frequency down by f0 and nothing up. The positive band lands at zero with its orientation intact, so the test tones appear at +10 kHz and −20 kHz, and the negative band moves to −600 kHz, where it no longer overlaps anything.
In hardware this is two mixers fed from one oscillator, one with the cosine and one with minus the sine, whose outputs are the in-phase (I) and quadrature (Q) channels. Seen in the helix, the mixer is a strobe at the carrier frequency: in a frame turning with the oscillator the carrier stands still, and what remains is the slow motion of the envelope and its phase. Moving the mixer frequency off the carrier sets the helix turning again at the difference frequency.
Step 6
A low-pass filter removes the copy at −600 kHz, which is the redundant half of the real signal's spectrum from step 2. What is left is the analytic signal of the echo shifted to zero frequency, and it occupies ±30 kHz, so a complex sample rate of 60 kHz holds it.
Counting real numbers shows that nothing has been gained or lost. 60 k complex samples a second are 120 k real numbers, the same as real sampling at twice the bandwidth; the bandpass window of step 3 needs 132 k because the band edges do not fall on multiples of the bandwidth, and sampling above the carrier needs 660 k. ApertureLab's simulator stores 75 k complex samples a second, 1.25 times the bandwidth.
Step 7
The real echo is the real part of the baseband signal put back on the carrier. Interpolating the baseband samples to a fine time grid, here with the ideal sinc interpolator, multiplying by e+j2πf0t and keeping the real part reproduces the voltage of step 1:
From 60 samples the reconstruction differs from the original by 145 dB below its peak. The record length sets that error, because the record cuts off the tails of the pulse, and the error falls further as the record grows. Shortening the record to 9 samples raises it to 74 dB below the peak. In both cases the limit is the truncated interpolator, not information lost in demodulation.
Step 8
Delaying a signal on a carrier by τ delays its envelope by τ and multiplies its baseband by a phase factor:
The baseband phase is therefore the carrier phase that the mixer took away, and for an echo τ = 2R/c, so the phase is −4πR/λ. It turns once for every half wavelength of range, 2.5 mm, while the envelope moves a fifth of a resolution cell. The magnitude says which cell the target is in; the phase locates it within the cell.
Step 9
A back-projection beamformer forms each pixel by summing, over every ping, the sample at that pixel's two-way delay τ. For the toy sonar the synthetic aperture is 2.79 m long and the delay changes across it by enough to turn the carrier phase through 129 rad from the center to the edge. By step 8 each baseband sample carries the factor e−j2πf0τ, so the sum only adds in step if every sample is first multiplied by its inverse:
Without the rotation the phasors from successive pings curl into a spiral, and the toy point target loses 15.7 dB. Rotating with the wrong sign doubles the phase error instead of canceling it and costs 18.7 dB in the toy. That is the error a convention mismatch produces: physics texts write a wave as e−iωt and engineering texts as e+jωt.
The rotation also has to use the frequency the data were demodulated with. The figure below shows the peak lost when the rotation frequency is off by a given percentage: a 1 % error is negligible, and a 10 % error costs 6 dB in the toy model and 7 dB in ApertureLab's beamformer.
ApertureLab's beamformer applies the same rotation to the 36-channel array, and the board shows its images with the rotation set each way, on a seabed simulated at 20,000 scatterers per square meter. With the rotation off, the point target loses 17.8 dB and the seabed's speckle smears into streaks along the track. With the wrong sign it loses 17.0 dB, within a decibel of no rotation; in the toy model the wrong sign costs 3 dB more than no rotation.
Delay-and-sum on the real passband signal, sampled at 3 MHz, forms the same image without any rotation, and its envelope peaks within 0.2 dB of the baseband image. The passband image oscillates once every half wavelength of range, 2.5 mm, which is the carrier, so its pixels must be finer than a quarter wavelength to represent it. Baseband processing does the same summation with the carrier handled exactly in the phase factor, which lets the samples be interpolated at the bandwidth rate rather than above twice the carrier frequency.
Method
Steps 1 to 9's live views are computed in the browser from a model of one echo: a compressed pulse with a Hann-weighted 60 kHz spectrum on a 300 kHz carrier, with time measured from the echo of a point at the reference range and the oscillator in phase with that echo. The toy aperture of step 9 places one phase center every 15 mm across the beam-limited aperture of a 30 mm element, tapers it with a Hann window, and back-projects baseband samples at 75 kHz with linear interpolation; the passband comparison samples the real echoes at 3 MHz. Its point focuses to 24.5 mm along-track, the 1.44λR/2L of a Hann-tapered aperture.
The board's images come from ApertureLab's point-scatterer simulator and time-domain back-projection beamformer, on the still-water scene of The surface of focus: the 36-channel array at 300 kHz with 60 kHz of bandwidth, a track 10 m above a flat sand bed, and the beamformer given the true path. Each image was formed on a 1.25 cm grid with the carrier rotation set as the control shows and every other setting unchanged. The point target's response is isolated by subtracting the image of the same scene simulated without it, which is exact because the imaging chain is linear, and is measured on a 2.5 mm grid; with the correct rotation it is 21 mm wide along-track, finer than the toy's because the beamformer integrates over 1.5 times the element's beam.
For the steps that lead up to a synthetic aperture, see Why synthetic aperture?; for the full imaging chain, including pulse compression and micronavigation, see Image formation.
References