Sinusoidal Frequency Estimation with Applications to Ultrasound
Sinusoidal Frequency Estimation with Applications to Ultrasound Sinusoidal Frequency Estimation with Applications to Ultrasound
Section 2.2. Single Tone Estimators 215−5N = 12N = 16N = 20N = 24CRLB−15MSE (dB)−25−35−45−55−65−10 −5 0 5 10 15 20 25 30SNR (dB)Figure 2.3. The MSE of the examined KWPA estimator as a function of the SNR, ω =0.75π.In [KNC96], Kim et al. proposed using a simple K-tap moving average filter to smoothirregularities and random variations prior to the frequency estimation as a way to reducethe SNR threshold. Such an averaging can be shown to lower the SNR threshold up to10 log 10 K dB. However, as such an averaging will severely restrict the allowed frequencyrange down to (−π/K, π/K], the method in [KNC96] is limited to signals with frequenciesnear zero. This is because the finite impulse response (FIR) averaging filter is essentiallya low pass filter. Herein, it is noted that the frequency content of the downshifted signal,y d (t), will satisfy such a restriction, and it is therefore proposed to instead form an averaged
Section 2.2. Single Tone Estimators 225−5ω = 0ω = 0.25πω = 0.5πω = 0.75πCRLB−15MSE (dB)−25−35−45−55−65−10 −5 0 5 10 15 20 25 30SNR (dB)Figure 2.4. The MSE of the examined UWPA estimator as a function of the SNR, N = 24.signal asy f (t) = 1 KK−1∑k=0y d (t + k). (2.2.25)Similar to (2.2.13), the adjacent phase difference of (2.2.25) can be formed as∆φ f (t) = arg [ y ∗ f(t)y f (t + 1) ] = ω f + u c (t), (2.2.26)where u c (t) is given by (2.A.9) for a general K (see Appendix 2.A for further details). It isworth noting that the noise process u c (t) will now be coloured due to the average filtering.As shown in [LM89], the SNR threshold behavior of the phase-based frequency estimatorsis affected by cumulative ±2π phase errors resulting from the effect of the additivenoise. This effect can be countered by introducing an outlier detection scheme. Recently, an
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Section 2.2. Single Tone Estima<strong>to</strong>rs 225−5ω = 0ω = 0.25πω = 0.5πω = 0.75πCRLB−15MSE (dB)−25−35−45−55−65−10 −5 0 5 10 15 20 25 30SNR (dB)Figure 2.4. The MSE of the examined UWPA estima<strong>to</strong>r as a function of the SNR, N = 24.signal asy f (t) = 1 KK−1∑k=0y d (t + k). (2.2.25)Similar <strong>to</strong> (2.2.13), the adjacent phase difference of (2.2.25) can be formed as∆φ f (t) = arg [ y ∗ f(t)y f (t + 1) ] = ω f + u c (t), (2.2.26)where u c (t) is given by (2.A.9) for a general K (see Appendix 2.A for further details). It isworth noting that the noise process u c (t) will now be coloured due <strong>to</strong> the average filtering.As shown in [LM89], the SNR threshold behavior of the phase-based frequency estima<strong>to</strong>rsis affected by cumulative ±2π phase errors resulting from the effect of the additivenoise. This effect can be countered by introducing an outlier detection scheme. Recently, an
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