In radio astronomy the signals being collected are typically much weaker than the noise and a lot of effort is put into building high sensitivity, stable receiving systems. The individual telescopes are big to collect as much light as possible and hence reduce integration times. One method that can improve signal-to-noise is interferometry, where signals from two or more telescopes are cross-correlated (in pairs). The signal from an astronomical source, received by each telescope will be preserved but the noise that is different at each telescope will be different and therefore suppressed in the correlation.
The following simulations are of audio data with similar properties to my data from the prototype recorder. Sampling is at 96 kHz giving a maximum audio frequency of 48 kHz. The initial assumption is that the microphones are co-located and therefore there is no delay between receiving the same signal.
These simulations compare the noise levels from a single microphone to that obtained by combining two of them. The data are combined in two ways:
- The first is a simple summation of the data streams in the time domain. This emulates a technique that is commonly used in the data collection process. Adding the two signals together doubles the signal strength (i.e. a 6 dB increase) and increses the noise by 3 dB for a 3 dB improvement in SNR overall (InvenSense Application note AN-1140).
- A technique used for radio interferometry is to cross-correlate the two data streams. This is done by multiplying the Fourier transform of one channel by the conjugate of the Fourier transform of the other.
The noise that is recorded from each microphone will contain a component generated by the microphone itself as well as noise generated elsewhere in the system that is common to both.
In this simulation two data streams are made that contain only random Gaussian noise with a fraction of that noise common to both that is varied. If all noise is common then there should be no difference between the single and two-microphone case. The signal noise floor has been calculated as a function of this fraction.
The following plot shows the predicted noise floor for a single microphone ("Left"), the sum from a pair of microphones and the cross-correlation of the two signals. As predicted, in the case where the noise from the two microphones is independant (i.e. fraction correlated noise is zero), the noise floor for the sum of two data streams is a factor of two (3 dB) higher. A cross correlation shows dramatic improvement in noise level. For example, data where 10% of the noise is common to both channels will result in a 10 dB decrease in noise.
We can compare these predictions with actual data from the prototype recorder. For this test, I wanted to measure the noise floor so the microphone pair was sandwiched between two pieces of acoustic foam[^1] and a two-second recording was made.
The following graph shows the power spectral density as a function of frequency for the two microphones, their sum and the cross-correlation. Unlike the simulated data, which assumes that noise is constant as a funtion of frequency, the microphones have a changing response across the band.
The spectral density plot shows very similar noise floors within the specified range but significant differences above that. It is however very clear that the noise floor of the cross-correlated data is much lower than the individual microphone signals.
[^1]: Which we originally bought to attenuate the sound coming from my son's drum kit!
The next graph shown the difference in power levels of the summed and cross-correlated data as a function of frequency. For the range of frequencies below 15 kHz, the 3 dB change is clearly seen. The improvement in noise for cross-correlated data is approximately 10 to 15 dB with an average of ~-12.5 dB or a factor of 0.056 or 6% correlated noise. In other words, over 90% of the internal noise can be removed using cross-correlation.
Signal to noise ratio
In the next set of simulations I combined the noisy data with a 10% common component with an input signal to estimate the difference in signal-to-noise betwen single and dual microphone configurations. The expectation is that the cross-correlation technique will provide the ability to detect much weaker signals than a single microphone or the sum of signals from two microphones.
A 15 kHz signal was inserted at varying intensity. The peak value was measured and compared to the noise level. To make the comparison clearer, the summed data have been expressed as an average (i.e. (Left+Right)/2). The following figure shows measured signal strength compared to input. As the signal gets weaker, the measured signal at 15 kHz becomes more and more dominated by the noise and the curve departs the input=output line. As expected, the average and cross correlation techniques perform better than a single microphone with the cross-correlated data the most sensitive.
The next plot shows signal-to-noise ratio (SNR) as a function of input signal strength and indicates the improvement in performance. If for example, the minimum acceptable SNR is 5, the minimum signal strength is
- 58 V/sqrt(Hz) for single mic
- 43 V/sqrt(Hz)for summed data
- 18 V/sqrt(Hz) for cross-correlated data.
In other words, for a SNR detection limit of 5, cross-correlated data is capable of detecting signals 3 times weaker than a single microphone and 2.4 times weaker than summed data.
In the next post I’ll show some real signal detections and see if SNR is improved as expected…