Case study: Audio SNR and filtering

A well-defined signal-to-noise ratio must include a filter. This example explains, what can happen, if it is neglected:
A SNR measurement without additional filter is repeated with “bad” and “good” measurement equipment.
Curiously, the “bad” equipment gives better results!

Introduction

Consider the following setup:
An audio signal is recorded with a microphone. The microphone adds “white” noise (other noise contributors are negligible), which is constant over all frequencies. The noise causes a deterioration in signal-to-noise ratio.
The task is to measure the SNR of the recorded signal.

The experiment is repeated twice:

Using an ordinary sound card at 44.1 kHz sampling rate (CD quality)
Using a high-end converter with 96 kHz rate

Case 1: SNR detection without filter

The simple measurement setup is shown below. It does not include a dedicated filter for SNR detection:

The only filter in the signal path is the lowpass filter of the sound card, namely the antialiasing filter of the analog-to-digital converter.

For 44.1 kHz sampling rate, the card's lowpass filter is not much wider than the audio signal:

The high-end soundcard uses a filter with twice the bandwidth:

The bandwidth is only limited by the wide frequency response of the high-quality soundcard. Microphone noise in the ultrasonic range passes through, and will deteriorate the SNR.
The measurement with the worse instruments measures better SNR!
Of course, the result shouldn't depend on the instruments at all. Therefore, a better solution is shown below:

Case 2: SNR detection using a filter

Now the setup includes a dedicated filter prior to the SNR-detection:

Used with the low-rate sound card, the filter limits some inaudible high frequency noise, and results in a small SNR improvement:

With the high-end card running at high sample rate, the bandwidth seen by the SNR detector is exactly the same as in the low rate case.
Consequently (all other factors equal), the SNR will be exactly the same as with the low-rate card:

Conclusion

Measuring SNR as outlined above requires an additional filter to get meaningful results, because otherwise the noise bandwidth is implicitly defined by the measurement instrument bandwidth.
In an audio context, SNR is usually not that critical. But in communications engineering, 2 dB of SNR can make a dramatic difference: bit error rate may change by a factor of 10.
Or, to quote a colleague: “(cellular phone) operators will kill for one dB”.

Comments? Questions?