Frequency content of discrete-time signals
During lecture, I attempted to illustrate, through music (a snippet of
Chinese opera), that when we sample a signal at some sampling
frequency fs, the only frequencies present in the
reconstructed signal lie in the range [-fs/2,
fs/2]. I took a piece of music from a CD (sampled at
44.1kHz) and subsampled it at 22.05kHz, 11.025kHz, 8kHz, 4kHz and
2kHz. According to sampling theory, the highest frequency components
that can be reconstructed for each sound file are half of each
sampling frequency. For example, for the music snippet with sampling
rate fs = 8kHz, we should not be able to hear any
frequencies greater than 4kHz; we saw that this is indeed the case,
by playing each subsampled piece of music in an mp3 player, and
manipulating different frequency bands in the equalizer of the mp3
player. As an example, an 8kHz sampled signal should sound exactly the
same for the two equalizer configurations pictured below, and,
indeed, it did.
A perceptible difference in music sampled at lower frequencies is that
it sounds muffled, or completely unrecognizable (as is the case for
fs = 2kHz). To listen for yourself, click on the
music sound files below.
Cantonese Opera (1 min.)
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Sampling rate
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Music files
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44.1kHz
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(mp3, 2.3 Mb)
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(wav, 10.1 Mb)
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22.05kHz
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(mp3, 2.3 Mb)
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(wav, 5 Mb)
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11.025kHz
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(mp3, 2.3 Mb)
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(wav, 2.5 Mb)
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8kHz
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(mp3, 2.3 Mb)
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(wav, 1.8 Mb)
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4kHz
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(mp3, 2.3 Mb)
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(wav, 948 kb)
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2kHz
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(mp3, 2.3 Mb)
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(wav, 476 kb)
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Last updated February 4, 2003 by Michael C. Nechyba