3.3 Periodogram-based estimation algorithms
3.3.6 Signal detection and false alarm rate
One algorithm able to solve this bounded, two-dimensional optimiza- tion problem is the Nelder-Mead simplex algorithm [75]10 (for one di- mension, [87] suggests a secant method which would not work for this two-dimensional problem). This algorithm requires initial values, which can either be the coarse indices (ˆn,m) or even fractional indices from aˆ previous quadratic interpolation for faster convergence.
Constant False Alarm Rate
Most radar systems are configured for aconstant false alarm rate(CFAR), and this solution is also chosen for the OFDM radar system. The pre- cise definition of the CFAR varies in literature (compare [98] and [97], for instance), which is why the following definitions are introduced:
A false alarm is the event where the target detector decides that there is a target at a range and relative speed which did not contribute to the received matrixFRx. The probability of a false alarm PFA is the probability that, during the processing of a single frame, one or more false alarms occur when only noise was present (FRx = Z). Finally, the false alarm rate (FAR) is the expected number of detections per processing of one frame, for the case that only noise was present.
This definition differs from other definitions of false alarm probability (and FAR) in several respects:
• The time base for the false alarm rate is the duration of one frame (as a comparison, [98] suggests the number of false alarms per sec-
ond for the FAR). This makes the results discussed here indepen- dent of the update rate.
• Clutter is explicitly not discussed in this context. The detection of an object that backscatters energy, but is not of interest for the application, thus does not count as a false alarm (an example would be the detection of a traffic sign in a vehicular context).
• Other systems, such as target tracking algorithms [138], might fur- ther process the output of the target detector (cf. Fig. 3.6), thereby possibly further reducing the false alarm rate.
In order to discriminate noise from signal power, a thresholdη is intro- duced, and the periodogram is subjected to a hypothesis test,
PerF(n, m)H≶0
H1
η, (3.90)
where H0 is the null hypothesis (no target is present) and H1 is the hypothesis that a target contributes to the amplitude of the given bin.
To calculate the false alarm probability, letZ denote the (random) am- plitude of any bin of PerF(n, m) when only noise is present (due to the whiteness of the noise,Z is i.i.d. for all bins).
The probability that any single bin of the periodogram exceeds the threshold is
pFA,bin := Pr [z> η] = Z ∞
η
fz(z|H0)dz= 1−FZ(η|H0) (3.91)
=e−
η σ2
N, (3.92)
wherefZ(z|H0) andFZ(η|H0) are the PDF and CDF, respectively, of the random variableZ. The exponential term (3.92) is the result ofZ being the magnitude-squared of AWGN with powerσN2, and thus exponentially (χ22) distributed.
To achieve a certain per-bin false alarm rate, solve (3.92) forη:
η=−σ2NlnpFA,bin. (3.93) The optimality of this detection method is discussed in [98, Chap. 15].
In order to achieve a specific false alarm probability, note that, for a non-zero-padded periodogram, the false alarm probability is
pFA = 1−(1−pFA, bin)N M. (3.94) Solving this forpFA, bin and inserting into (3.93) yields
η=σ2Nln(1−(1−pFA)N M1 ). (3.95) If the requirement is a certain FAR, which is calculated by
FAR =N M pFA, bin, (3.96)
the threshold is set by
η=−σ2NlnFAR
N M. (3.97)
The choice between (3.95) and (3.97) is determined by a trade-off: Fixing a FAR≥1 results in a lower threshold than using (3.95), and thus higher detection probability, but increases the burden on post-processing com- ponents downstream, as it also increases the number of false alarms.
When using the cropped periodograms as described in Section 3.3.2, the factorN M is replaced by Nmax(2Mmax+ 1) in (3.95) and (3.97). How- ever, when zero-padding is used, these values are not increased by the interpolation factor. It is true that if there are more bins in the peri- odogram due to zero-padding there are potentially more bins to cause
false alarms. However, zero-padding does not add information to the periodogram, it merely reduces quantization error, and adjacent bins in zero-padded periodogram are correlated. A single peak in a non-zero- padded periodogram will thus cause multiple elements of a zero-padded periodogram to lie above the threshold, but as they are contiguous, the detection algorithm (see Section 3.3.7) will only detect this peak once.
Noise power estimation
Usually, the noise power σN2 is not known at the receiver. To still be able to specify a threshold, the noise power may be estimated from the periodogram by averaging over those bins which do not contain a target.
As this happens before the target detection, it is unclear which bins correspond to targets.
The solution is to rely on the correct parametrization of the OFDM radar system: As discussed in Section 3.3.2, there is a maximum index (Nmax) after which no more peaks should appear. By averaging over one or more rows beyondNmax, a maximum likelihood estimate forσ1N can be found by
ˆ
σ2= 1 MPerK
XK k=1
MPer
X
m=0
PerF(NPer, m), (3.98) where K is the number of rows over which to average. Unless MPer is very small, a value of 1 or 2 forK is sufficient.