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Intralab Memorandum INTRALAB ME MORANDUM To: Distribution From: Joe Paradiso Date: 31-Dec-1990 Subject: Sonar Application of a Quadrature-Sampled Interpolation Beamformer Memo #: EJB 90-191 Copies: Abstract This memo discusses the application of quadrature sampling to acoustic beamforming. Since they require the sampling frequency to be at least twice the signal bandwidth, quadrature techniques enable narrowband signals to be sampled at frequencies much lower than the carrier's Nyquist limit. Second-order sampling techniques are used to obtain in-phase and quadrature components from distributed hydrophone elements. A MATLAB simulation of a quadrature- sampled interpolation beamformer has been developed, and simulation results are presented for both a line array and the arc-shaped WQS1 receiver. A design is suggested for a practical hardware implementation of the second-order quadrature sampling technique, and issues are discussed with relevance to beam formation via DSPs. 1) Quadrature Sampling Active sonars often employ highly narrowband signals. In the beamforming application illustrated here, essentially a pure sinusoid is emitted, and the return signal is broadened somewhat by Doppler scattering, dispersion, etc. (but remains narrowband). Any bandpass signal can be represented by a sum of in-phase and quadrature components: ω ω 1) x(t) = xI (t) cos 0t + xQ (t) sin 0t The in-phase [xI ] and quadrature [xQ ] signals represent a slowly-varying complex envelope which is applied to the high-frequency carrier at ω0. Under quadrature sampling [1], the signal can be sampled at the Nyquist cutoff of the complex envelope (i.e. at the signal bandwidth) rather than at twice the carrier frequency (2f0 = ω0/π), yielding an appreciable reduction in sampling rate for narrowband signals. Beamformers, in particular, can require significant oversampling in order to form synchronous beams (where beamforming delays are attained by shifting signals forward and backward by discrete sample bins). Oversampling to 2 or 3 times Nyquist rates are common in beamforming applications, hence an even greater comparative sampling rate reduction can be attained when using quadrature sampling combined with an interpolation filter, as will be discussed in the next section. Demodulation cos ω t 0 Pick Lower Input Sideband Complex Sampling X(t) Low Pass X (t) Filter I H(f) Low Pass Filter X (t) H(f) Q sin ω t Sampling 0 Frequency fs Figure 1: Demodulation Sampling of Quadrature Signals 2 A complex demodulation is the standard approach used to obtain quadrature samples, as illustrated in Fig. 1. The input signal is multiplied by the output of a quadrature oscillator running at the demodulation frequency (generally close to the carrier frequency, to baseband the modulated signal). The sine and cosine products are low-pass filtered to remove the sum frequencies. The downshifted difference frequencies are digitized at a rate (fs) greater than signal bandwidth, yielding a pair of in-phase and quadrature samples. In some cases [2], the modulation frequency may be displaced from the carrier (f0) and moved outside the signal bandwidth. This is done to prevent components of the input signal from translating down to DC and adding a bias into the output that affects the dynamic range of the analog-to-digital converters (i.e. all signal components will be purely AC). In these situations, the effective signal bandwidth is increased; i.e. the sampling rate must be augmented by the displacement of the modulation frequency from the carrier in order to avoid aliasing. Input nl x (t) X(t) (-1) Q Sample Sampling Frequency 2f f = 0 s l Delay: 1/(4f0 ) Sample nl (-1) x I (t) Figure 2: Second-Order Method Approximating Quadrature Sampling If the input signal is highly narrowband, as it is in the sonar applications considered here, quadrature sampling may be performed via a second-order technique [1]. This method can be much simpler to hardware-realize; as seen in Fig. 2, the analog multipliers and filters implicit in the -1 demodulation of Figure 1 are not needed. In-phase signals are sampled at a period of ∆ = l (2f0) , where l is an integer that must be selected such that ∆ remains less than W-1, the inverse of the input signal's bandwidth, to avoid aliasing. The quadrature samples are taken after the signal is delayed by -1 α, where α = (4f0) + K∆, where K is an integer. The simplest case is to pick K to be zero; here the 3 quadrature samples are taken after a quarter-period of the carrier has elapsed, relative to the in-phase -1 samples. The sampling frequency fs = ∆ is again well below the carrier frequency here, since we are sampling the narrowband complex envelope of the signal. The approximation implied in the second-order technique is that the in-phase and quadrature samples are simultaneous. This, of course, is not realized; they are mutually separated by the interval α. This argument leads to a relation [3]: 2) xQ(n∆ - α) = xQ(n∆[1 - α/∆]) = xQ(n∆[1 - 1/(2l )]) = xQ(n∆[1 - W/(4f0)]) If the undersampling factor (l) is sufficiently large, the error (i.e. time displacement of the quadrature sample) will be small. If l is set to its maximum value (as it usually is, resulting in Nyquist sampling; i.e. ∆ ≈ W-1), the rightmost equality holds. If the bandwidth of the input signal is much less than its carrier frequency (i.e. highly narrowband assumption), the simultaneity error of Eq. 2 is negligible. The quarter-period delay could be directly accounted for by using an interpolation filter to digitally upsample to 4f0, and phase-shifting the quadrature signal by a single bin. This is not needed in practice [3], however, as the response of the sonar systems considered here should be sufficiently narrowband for the coincidence assumption to be valid. The interpolation filters used in the beamformer (discussed below) need not oversample to such a high frequency to obtain satisfactory beam steering. The in-phase and quadrature samples obtained from the second-order sampling technique must both be multiplied by a sign-changing factors of (-1)nl, where n is the sequential sample bin (i.e. at time n∆), and l is defined above. Note that the scheme in Fig. 2 involves delaying the sampling gate, as opposed to delaying the signal itself (as performed in [1] & [3]). This results in an effective interchange of in-phase and quadrature signals; i.e. the quadrature signal is taken earlier under this sign assumption. The input signal must be appropriately bandlimited when quadrature sampling at a lower frequency, as in Fig. 2. If any baseband frequencies are present on the input signals (i.e. AC hum, surf noise, fish sounds, etc.), they will alias into the digitized samples. In such a case, the input signal must be bandpass filtered before being input to the second-order sampler. 4 Shading Coefficients a1 τ1 aNE τ NE Figure 3: A Standard Beamformer 2) Interpolation Beamforming The layout of a standard beamformer is shown in Fig. 3. An array of hydrophones (with optionally weighted outputs an that equalize signal levels and/or "shade" the array) produces signals upon the arrival of a plane wave. The directional response of the array is steered by introducing an adjustable delay τn into the sensor signals, which are then summed together. A large-amplitude constructive sum occurs when all hydrophones produce an in-phase (correlated) signal. Otherwise the signals decorrelate and interfere, hence the magnitude of the beamformer output is considerably smaller. The delays, derived from the geometry of the hydrophone array, are adjusted such that the sensor inputs to the beamformer are in phase when the plane wave arrives from the desired direction (i.e. see Ref. [4]). Multiple beams projecting in several directions may be formed simultaneously from the same set of hydrophones by fanning the sensor data into beamformers with different delays. Since the processes in the "beamformer" box of Fig. 3 are now usually performed digitally, several beams can easily be constructed on the same set of sensor data samples (in quasi-real-time; i.e. during the wait between successive sonar pings) without introducing redundant hardware. The ideal line-array beamformer can be shown [4] to have the response: sin n γ sin θ - θ 3) b(θ) = e 0 0 ne sin γ0 sin θ - θ0 5 γ 1 ω∆ Where: 0 = /2( x)/c ω = Frequency of received signal ∆x = Line element spacing ne = Number of elements in array c = Speed of sound in water θ0 = Beam Steering Angle This function is very familiar; it has a large central maximum (at the beam steering angle θ0), off- maximum low-level ripple, and grating lobes at large angle. In order to obtain synchronous beams over a reasonably broad range of steering delays, a digital beamformer must sample the signal sufficiently fast (frequently 2 to 3 times over the Nyquist limit). Synchronous beams occur when the element-to-element delay can be defined exactly in terms of an integer number of sampling periods ∆, thus the delays can be realized after the signal has been digitized by shifting the samples appropriately by the corresponding number of bins before summing them in the beamforming operation. If non-synchronous beams are formed, the delay is rounded to the nearest integral multiple of ∆, and the samples are similarly shifted and summed. In the non-synchronous case, however, this truncation causes the steering delays to be inexact, hence the beamformer looses precision, and the sidelobes & ripple grow in relative magnitude. This is demonstrated in Ref. [5]. The need to digitize at a sufficiently high rate to avoid this difficulty can significantly drive the hardware and data flow designs.
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