STUDIO

SINE WAVE1 2 3 4 5 6 7FREQUENCYSQUAREWAVE1 2 3 4 5 6 7disappear as you pass through the 50% point. Thenotch is quite sharp and only slight variations inthe duty cycle can cause significant amounts ofthe second harmonic to be in the signal. Thesecond harmonic is important because althoughthe 21 kHz third harmonic might reasonably befelt to be outside the normal hearing range, the14 kHz second harmonic should certainly bedetectable by most listeners.Verifying the harmonic purity of thesquarewave using a spectrum analyser hassimilar problems to those with the sinewave -theanalyser distortion can contribute to the harmoniccontent. Equally the noise floor of the analysercan mask a second harmonic, which is present inthe squarewave but below the noise of theanalyser. Even if the second harmonic is belowthe noise level in the original signal it has a fixedlevel and frequency, as opposed to thesurrounding noise -perhaps providing enoughclues for the ear to detect it -much as radioamateurs are able to work with Morsetransmissions with a negative signal to noiseratio: the time correlation gives away thepresence of the tone in the noise. This effect alsooccurs with the sinewave, so averaging of the92 Studio Sound, June 1989FREQUENCYspectrum display is needed to enhance thecorrelated frequencies and suppress the noisecomponents.We seem to have an escalating set ofrequirements for our basic equipment. The twosound sources need to provide a sinewave andsquarewave with as near perfect spectral purityas possible and to verify this we need a spectrumanalyser whose performance is as good if notbetter than the waveform generators. Oncesatisfied that the raw sounds are as near perfectas possible, the next stage is to couple them tothe listener.TransmissionIn order to hear the sinewave and thesquarewave, they need to be amplified andconverted into sound waves suitable for receptionby the human ear. The amplifier must not addany distortion to the sounds as this would addharmonics thereby removing some of the intrinsicdifferences -the second and third harmonics of the7 kHz fundamental being the most important inthis case. Obviously a high quality amplifier isneeded, with low noise and distortion.Note that we are not considering using a digitaltransmission medium as an intermediate stage -Iam staying in the analogue domain for thisdiscussion. Any digital coding or processing of thesignals would introduce bandwidth limitations,changes in noise floor and distortions, whichwould further degrade the purity of thewaveforms.More problematical than the amplifier is theloudspeaker to which the amplified signals areconnected. No speaker is perfect -a typical high -quality loudspeaker has a distortion of severalpercent -and so this will upset the carefullygenerated `pure' waveforms. The resultingsinewave will contain harmonics other than justthe fundamental, and the squarewave will containharmonics other than the fundamental and oddharmonics. The distorted sound waves will thenbe detected by the listener's ear and analysed.The distortion is particularly unfortunate sincethe missing second harmonic, which should beabsent from each waveform, could now be presentin both. The task of verifying the detection of the21 kHz component has been negated by the lackof certainty as to the presence or absence of anylower harmonics -in particular the 14 kHz secondharmonic.Despite careful attention to detail, we havefallen at the last hurdle! The sine andsquarewaves, having been checked rigorously forcorrect harmonic content, have been corrupted bythe loudspeaker. With any significant distortionpresent, we cannot verify that the sinewavecontains just the fundamental and thesquarewave no second harmonic, and so it looksas though we cannot prove anything with theexperiment, since we could be proving that theear is capable of detecting the 14 kHz secondharmonic.The experimentTo confirm the above, I carried out some tests onan informal basis. I used standard laboratoryfrequency synthesiser sources for the sine andsquarewaves, using a high -quality loudspeakerand amplifier. All three subjects tested were ableto reliably detect the difference between `sine' and`square' -waves up to 9 or 10 kHz. One of thesubjects commented that detecting the differencewas easier at the higher frequencies! Time andresources did not allow more formal testing undercontrolled conditions but the results wereconsistent and repeatable.ConclusionI must admit that I was originally sceptical aboutbeing able to detect a difference between asinewave and a squarewave at 7 kHz. The pointsdescribed above convinced me that the problemswere more to do with what was being measuredby the experiment. Assuming that the source andamplification are as near perfect as they can bemade, then the listener is probably detecting thedifference in the distortion of the loudspeaker forthe two signals. Using anything less than idealsignals under perfect experimental conditionswould muddy the water even more. I concludefrom this that the experiment as described doesnot confirm or deny the hypothesis that the earcan hear the third harmonic of a 7 kHzsquarewave.Acknowledgement is made to the Director ofResearch and Technology of British Telecom forpermission to make use of the informationcontained in this paper.