STUDIO

WHATARE WEMEASURING?Recent discussion on the ear's range ofhearing throws some question over thechoice of sampling rate for digitalrecording. Martin Russ considers the`7 kHz problem' at which point the earcan detect the difference between asquarewave and a sinewaveOne of the most interesting discussionswhich arose at the 1988 DigitalInformation Exchange concerned therange of hearing of the human ear.The choice of sampling rate for CDs and DAT wasfelt by some people to be too low and a higherrate was proposed as a means of accuratelycapturing the so- called `superharmonics' of audiosignals. Part of the justification for this was basedupon the apparent audible difference between a7 kHz sinewave and a 7 kHz squarewave. Thefirst component of the squarewave above thefundamental is the third harmonic at a frequencyof 21 kHz -well above the usually stated range ofhuman hearing. Because it is apparently possibleto hear a difference between the square and thesinewave, this was put forward as a proof that theear is capable of discerning tones higher than20 kHz -and thus the normal digital samplingrate of 44.1 kHz was put into question.Such an argument polarises the protagonistsinto two opposing camps. Electronics engineerswill probably accept the published data about theear and so reject the idea that such`superharmonics' exist, while audio engineers areconfident that there is something different abouta digitally coded signal and more specifically, thatis it not easily expressed in terms of THD, SNRor any other common measure of performance.This article will look at the '7 kHz problem'mentioned above and see if it does offer anyindication as to the actual frequency response ofthe human ear. It will not discuss the human earand its method of operation but will look at thephysics and electronics aspects of the problem,more within the author's sphere of knowledge.The problemCan you hear the difference between a 7 kHzsinewave and a 7 kHz squarewave? Morespecifically, is the ear capable of hearing the21 kHz component of the squarewave? In order toanswer this we need to look in detail at exactlywhat the two sound sources will actually looklike, rather than make assumptions about them.The sinewave seems simple enough -a singlefrequency at a specified level. Sinewaves are incommon use for aligning instruments in manylaboratories and studios, and are often referred toas a `pure' tone. Unfortunately the actuality israther different from the theory. A typicalsinewave will contain noise, other harmonics andwill have some frequency variation. Noise is afundamental limitation on any physical system -for exactly the same reasons that mixing deskshave a noise floor, so the sinewave signal willalso have an equivalent noise floor. Otherharmonics are often present in sinewaves becauseof the method of generation -filtering and shapinga square or triangular waveform may removemost of the frequencies above the fundamentalbut only very expensive and sophisticatedsinewave sources manage to get all this additional`clutter' below the noise level. The frequencystability of a sinewave is also limited by themethod of generation -and again highperformance is related to high cost.Verifying a sinewave source can also be aproblem. Connecting a typical sinewave into aspectrum analyser will probably show only asingle frequency but this depends on the noisefloor of the analyser, since if it is higher thanthat of the sinewave source, then any harmonicswill be masked by the analyser noise. Anydistortion in the input stage of the analyser willappear as extra harmonics in the sinewave, andso the presence of harmonics on the display mightnot accurately reflect the true harmonic content ofthe sinewave. Measuring the frequency stabilityof the sinewave source requires an accuratetimer -at which point the same sort ofuncertainty problems come forward again.The squarewave is also conceptually simple -awaveform with two states that occur for half thetime. Squarewaves are commonly found in digitalequipment and form a useful source for testingthe response of loudspeakers. The problem withsquarewaves is that they are impossible toachieve. A true squarewave has equal timeperiods for the two states and takes no time at allto change between the states. All real -worldsquarewaves take a finite time to move betweenthe two states and so are really trapezoidwaveforms. The slope between the two states andthe ratio between the two times (the duty cycle)both affect the harmonic content of thesquarewave -most importantly, they can affect thepresence or absence of the second harmonic.The second harmonic for a 7 kHz squarewave isat 14 kHz, and is present when the ratio betweenthe two states is not exactly one to one and alsowhen the time taken to change states is greaterthan zero. You can hear the effect on any soundsynthesiser that provides control over the dutycycle of the rectangular waveform. As you alterthe duty cycle towards 50% you can hear thesecond harmonic (an octave above thefundamental) drop in amplitude and almost91