Sample Rate Trade-offs

“Will my PA system sound cleaner if I run my mixer at a 96 kHz sample rate instead of 48 kHz?” The surprising answer is “No.”

Note: Grammatical errors in this article are proof that it was human-generated.

There. I’ve done it. I have challenged one of the Holy Grail tenets of digital audio and may have alienated a large contingent of the audio community. Before convicting me of Irrelevance, to await banishment to the Isle of Obsolescence please hear me out.

“More is better” – right? While there are times when that is true, it isn’t always the case in audio. A good example is the sample rate (SR) in an analog-to-digital converter (ADC). All digital products with analog inputs and outputs (I/O) have them.

It has not been long since the use of a 48 kHz SR was barely possible due to limits on processing power, data transport, storage, etc. Today’s ADCs can sample at many MHz (millions of hertz) and then decimate the SR to a practical value.

It is commonly believed that increasing the SR has the same effect on the audio signal as a higher pixel count in a digital camera. In the latter, the entire image is viewed in greater detail. Does a higher SR in a digital audio product (e.g 96 kHz vs. 48 kHz) capture the instruments and voices in greater detail, resulting in higher fidelity playback? Many claim to hear it. But is it true or is it the placebo effect?

Figure 1 – Top: A digital photo at 3 resolutions (dots-per-inch). More pixels yield higher resolution. Bottom: One-half cycle of a 1 kHz sine wave, sampled at 3 different SR. The tiny dots are the samples. (Click to enlarge).

Nyquist-Shannon Sampling Theorem
According to the Nyquist-Shannon sampling theorem, the required sample rate to fully resolve an audio waveform must be slightly higher than two times the highest frequency that it contains. If we accept that the high frequency limit of human hearing is about 20 kHz, the required sample rate is slightly higher than 40 kHz. This justifies the default sample rate of 48 kHz for many digital audio devices. One-half the SR is called the Nyquist frequency, and it represents the approximate high frequency limit of the device. Figure 2 shows the choices of SR (and resultant Nyquist frequency) for a hypothetical digital mixer.

Figure 2 – For some digital audio products, the SR can be selected by the user.

So, selecting the SR sets the Nyquist frequency, which represents the upper limit of the devices frequency response. I produced a summary graphic to serve as a reference (Figure 3). There’s a lot there, so study it carefully. It shows the border between the analog and digital domains as determined by the SR. The ultimate objective is the production of high fidelity analog audio (the green-ish region) using digital audio technology (the red-ish region).

Figure 3 – The frequency spectrum of interest when considering the effect of the sample rate on the audio signal.

Trouble at the Border
From Figure 3, the sample rate determines the Nyquist frequency, which determines the upper limit of the device’s audio bandwidth. Doubling the SR doubles the Nyquist frequency. A 48 kHz SR extends the analog audio bandwidth to nearly 24 kHz. In musical terms, this captures all of the 16 kHz 1/1-octave band. A simple conclusion is that each SR doubling adds an additional octave of audio bandwidth.

This “bandwidth extension” is the main benefit from increasing the SR. A pure tone (sine wave) between 20 Hz and 20 kHz does not benefit from a higher-than-48 kHz sample rate. For that reason you don’t get a “tighter” kick drum from a higher SR unless the kick drum has harmonics that extend beyond 20 kHz. Even then, the playback system must be capable of reproducing the harmonics at the listener, who could be tens or hundreds of feet from the loudspeaker.

I performed some measurements on a digital signal processor (DSP) to demonstrate the effect that SR selection has on the HF limit. Figure 4 shows the overlaid frequency response magnitudes for three common SR. The impulse response (IR) for each is inset.

It is an unfortunate reality that steep cutoff filters producing ringing in both the time and frequency domains (also Figure 4). This has always been true, but is now more common due to the availability of “brick wall” audio filters. The ringing can be reduced by special processing techniques, but it is always present to some degree. It is not necessarily a byproduct of inferior parts or design.

Figure 4 – The overlaid frequency response magnitude from 1 kHz – 100 kHz of three sample rates for a popular DSP. The IR for each is inset. The time span (800 μs) is the same for each IR.

1. Ringing can be reduced by increasing the SR.
2. The ringing for all three SR fall outside of the audible frequency range for humans (the vertical dashed line).
3. The frequency response to the left of the dashed line is identical for all three SR.