![]() I capture many wideband signals and RF scenes, and am supplied with such by others as well, and I have plenty of storage, since it is cheap and readily available. For a noisy signal, though, you also need to process enough data to overcome the effects of noise on the spectral correlation estimate. How do I decide if they’re too short or otherwise might lead to low-confidence analysis results?įor a noiseless random signal with cycle frequency, you’ll want to process a data block with length that is many multiples of to ensure you can see that periodicity. If you could detect such occurrences in your spectrum, you could then discount high coherence estimates that involve the spectral components near the detected edges/corners. Derivatives are tricky when dealing with noisy or random signals though. You might consider building into a power spectrum estimator the ability to detect spectral edges or corners or the like, perhaps using a derivative or similar. For a sine-wave component, the spectrum for the frequencies near the sine-wave frequency will inevitably reflect the tone’s energy, not just the noise or other continuous-spectrum signal components that lie there. For a step-function-like spectrum (such as a square-root raised-cosine QAM signal with zero roll-off), the edges of the spectrum cannot be perfectly estimated. They possess features that have spectral widths of zero, and so any spectrum estimate that possesses a finite effective spectral resolution cannot resolve them. But there are spectrum features that are inherently unresolvable, such as unit-step functions and additive sine-wave components. You’d have to know the true PSD of the signals in the processed data in advance. I don’t think it will be possible to automatically decide whether a PSD estimate is well-resolved. I’m guessing that it depends on the application requirements, but do you have any rules of thumb? I’m just wondering what your thoughts are on how to automatically decide whether a PSD is “well resolved” or not. If the PSDs are not well resolved, or contain zeros, the coherence quotient can be numerically unstable or erroneous. The tricky part of estimating the coherence involves good selection of the estimator for the denominator PSDs. In practice, of course, the numerator and denominator of the coherence are estimates corresponding to finite-duration data blocks. The data-block length is samples and the frequency resolution (width of in the FSM) is set to (one percent of the sampling rate, which here is ) for both the numerator spectral correlation function and the denominator PSDs. Estimated spectral coherence magnitudes for a rectangular-pulse BPSK signal with a bit rate of 100 kHz and a carrier frequency offset of 50 kHz. Here is an FSM-based estimate of the coherence for our rectangular-pulse BPSK signal: Figure 1. Therefore, the coherence lies in the closed unit disk in the complex plane. Since the coherence is a valid correlation coefficient, its magnitude will be less than or equal to one, and since the involved random variables are complex-valued, in general, so is the spectral correlation function and the coherence. Which is a normalization of the conjugate spectral correlation function. A similar argument holds for the conjugate spectral coherence, We call this function the spectral coherence. If the limit of the quotient exists, then the correlation coefficient is given by Now, as, the numerator of converges to the spectral correlation function and įor finite, then, we can form the correlation coefficient Here we are assuming that the two variables have zero means, which is always true when there is no additive sine-wave component in the data with frequency. To form the correlation coefficient, we need the variances for these two random variables, ![]() Which is the correlation between the two random variables and. For each finite, we can consider the quantity Where the expectation operator can be either a time average or the ensemble average used in a stochastic-process formulation. Now consider the spectral correlation function, ![]() So the correlation coefficient is the covariance between and divided by the geometric mean of the variances of and. Where and are the mean values of and, and and are the standard deviations of and. Let’s start with reviewing the standard correlation coefficient defined for two random variables and as See the posts on the strip spectral correlation analyzer and the FFT accumulation method for examples. It deserves its own post because the coherence is a useful detection statistic for blindly determining significant cycle frequencies of arbitrary data records. In this post I introduce the spectral coherence function, or just coherence.
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