Signals and Noise

Learn It!

Goal

The maximum frequency of blue whale songs is less than 50 Hz. Based on this information,
what is the minimum amount of memory required to store 1 hour of a blue whale song if it
is stored as a single channel of uncompressed audio using 16 bit quantization? (Use \(1 MB = 10^6 B\))

Part 1

Arranging terms

This problem suggests that we use dimensional analysis. There are many "this per that" units, and few directly applicable
equations that we know about this. We know that sampling rates will come into play at some point based on the context
of the question.

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1A

Dimensional Analysis

\(f_{Nyquist}=2 \cdot f_{max}=100Hz\)

Since this is digital audio, we know that we will need to be sampling it, and the
sampling rate needs to be at least the Nyquist limit. After this point we don't really care what the original
signal frequency.

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!

Limitations of dimensional analysis

When doing calculations like this, little unitless factors of 2 like the one here
often can get lost. Dimensional analysis is a good problem solving strategy, but it doesn't tell us
everything. It is often a good place to start, even if it doesn't solve the problem by itself

Sampling Rate: \( \frac{100 samples}{second}\)

Time: 1 Hour

Samples: \( \frac{16 bits}{sample}\)

Convert: \(\frac{3600 seconds}{hour}\)

Convert: \( \frac{10^6 Bytes}{MB}\)

Convert: \( \frac{8 bits}{Byte} \)

Time: 1 Hour

Samples: \( \frac{16 bits}{sample}\)

Convert: \(\frac{3600 seconds}{hour}\)

Convert: \( \frac{10^6 Bytes}{MB}\)

Convert: \( \frac{8 bits}{Byte} \)

Let's take inventory of the things we know, being careful to note all the units.
Since multiple units for time and data are mentioned in the problem, it is a good idea to write down
the conversion factors. Remember, MB represents

*MegaBytes*, Mb is*Megabits* \( \frac{16 bits}{sample}\)

Our final answer has dimensions of data, so we should start with a physical quantity that has bits
in the numerator.

\( \frac{16 bits}{\color{pink}{\cancel{sample}}} \cdot \frac{100 \color{pink}{\cancel{samples}}}{second} \)

Multiply by something that has samples. Samples cancel.

\( \frac{16 bits}{\color{pink}{\cancel{sample}}} \cdot \frac{100 \color{pink}{\cancel{samples}}}{\color{blue}{\cancel{second}}}
\cdot \frac{3600 \color{blue}{\cancel{seconds}}}{hour}
\)

Multiply by something that has seconds. Seconds cancel.

\( \frac{16 bits}{\color{pink}{\cancel{sample}}} \cdot \frac{100 \color{pink}{\cancel{samples}}}{\color{blue}{\cancel{second}}}
\cdot \frac{3600 \color{blue}{\cancel{seconds}}}{\color{black}{\cancel{hour}}} \cdot 1 \color{black}{\cancel{hour}}
\)

Multiply but number of hours. Hours cancel.

\( \frac{16 \color{green}{\cancel{bits}}}{\color{pink}{\cancel{sample}}}
\cdot \frac{100 \color{pink}{\cancel{samples}}}{\color{blue}{\cancel{second}}}
\cdot \frac{3600 \color{blue}{\cancel{seconds}}}{\color{black}{\cancel{hour}}}
\cdot 1 \color{black}{\cancel{hour}}
\cdot \frac{byte}{8 \color{green}{\cancel{bits}}}
\)

Convert bits to bytes. Bits cancel.

\( \frac{16 \color{green}{\cancel{bits}}}{\color{pink}{\cancel{sample}}}
\cdot \frac{100 \color{pink}{\cancel{samples}}}{\color{blue}{\cancel{second}}}
\cdot \frac{3600 \color{blue}{\cancel{seconds}}}{\color{black}{\cancel{hour}}}
\cdot 1 \color{black}{\cancel{hour}}
\cdot \frac{\color{red}{\cancel{byte}}}{8 \color{green}{\cancel{bits}}}
\cdot \frac{MB}{10^6 \color{red}{\cancel{bytes}}}
\)

Convert bytes to megabytes. Bytes cancel. The only unit remaining is MB, which is what we want!

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?

Does this expression make sense?

Taking more samples would increase the size of the file, so the sampling rate (100) goes up top.

Recording the data at higher precision should also make the file bigger, size of each sample (16) goes on top too

Longer songs use more memory, the length of the recording (1) should also go in the numerator.

Recording the data at higher precision should also make the file bigger, size of each sample (16) goes on top too

Longer songs use more memory, the length of the recording (1) should also go in the numerator.

★

\(\frac{16 \cdot 100 \cdot 3600 \cdot 1 }{8 \cdot 10^6}=0.72MB\)

Multiply everything together and we get less than a Megabyte.

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!

But a 3-minute MP3 is bigger than that!

Indeed, even a

*much shorter, compressed*MP3 typically has a file size of at least 1 or 2 MB. The trick is that 50Hz is a REALLY low frequency, and low frequencies need very few samples. Humans can hear sounds up to about 20,000Hz, and some instruments (like cymbals) create frequencies throughout that entire range. By comparison the range up to only 3400Hz was used for most telephone service during the last century. The 3000Hz-20000Hz range of high frequencies is mostly for abrupt sounds like cymbals and a few human speech sounds like 'sh' and 'psst'.