When you use sampling to produce a waveform, a side effect is caused when the sampling rate "beats" or combines with the frequency you wish to produce. This produces two additional frequencies, one at the sampling rate plus the desired frequency and the other at the sampling rate minus the desired frequency. This phenomenon is called aliasing distortion. Aliasing distortion is eliminated when the sampling rate exceeds the output frequency by at least 7 KHz. This puts the beat frequency outside the range of the low-pass filter , cutting off the undesirable frequencies. Figure 5-5 shows a frequency domain plot of the anti-aliasing low-pass filter used in the system. /|\ | 0db |________ filter | \ response | \ | \ | \ filter passes all frequences below about 5kHz. | \ -30db |_____________\_____ ___________________________________________\ | | | | | | / 5kHz 10kHz 15kHz 20kHz 25kHz 30kHz Figure 5-5: Frequency Domain Plot of Low-Pass Filter Figure 5-6 shows that it is permissible to use a 12 KHz sampling rate to produce a 4 KHz waveform. Both of the beat frequencies are outside the range of the filter, as shown in these calculations: 12 + 4 = 16 KHz 12 - 4 = 8 KHz /|\ filter 12kHz sampling frequencey | response | 0db |________ | | \ Diff. | Sum | | \ | | | | | \ | | | | | \ | | | | 4kHz| \ | | | -30db |_______|_____\_|___ ___|_______|_______________________________\ | | | | | | | / 5kHz 10kHz 15kHz 20kHz 25kHz 30kHz | | desired output frequency Figure 5-6: Noise-free Output (No Aliasing Distortion) You can see in Figure 5-7 that is unacceptable to use a 10 KHz sampling rate to produce a 4 KHz waveform. One of the beat frequencies (10 - 4) is within the range of the filter, allowing some of that undesirable frequency to show up in the audio output. /|\ filter 10kHz sampling frequencey | response | 0db |________ | | \ Diff. | Sum | | \ | | | | | \| | | | | \ | | | 4kHz| |\ | | -30db |_______|___|_\_____|_______|___________________________________\ | | | | | | | / 5kHz 10kHz 15kHz 20kHz 25kHz 30kHz | | desired output frequency Figure 5-7: Some Aliasing Distortion All of this gives rise to the following equation, showing that the sampling frequency must exceed the output frequency by at least 7 KHz, so that the beat frequency will be above the cutoff range of the anti-aliasing filter : Minimum sampling rate = highest frequency component + 7 KHz The frequency component of the equation is stated as "highest frequency component" because you may be producing a complex waveform with multiple frequency elements, rather than a pure sine wave.
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