SIMS-201

Characteristics of Audio Signals

Sampling of Audio Signals

Introduction to Audio Information

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Overview

Chapters 10-11 Background on audio signals Period, frequency and amplitude Audio signal components Sampling rate Undersampling and Oversampling Reconstructing audio from samples

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Background on Audio Signals

Sound is created by the motion of air particles in space (by applying mechanical force to air – ex. dropping a book on the floor will expand energy. Energy will move air around, which motion is perceived as sound.)

The changes in air pressure can be measured and recorded, for example by using a microphone (converts the air motion into an electrical signal)

We recall the “hello” wave of human voice:

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Properties of audio signals Audio signals have frequency components that are

complex In other words, most audio signals are made up of

several frequencies, combining to make the sound we recognize

The standard for human voice is taken to vary from about 100 Hz to 3000 Hz

Piano: Concert A above Middle C is 440 Hz Lowest audible frequency for humans is around 20

Hz (A low, rumbling bass note)

Highest audible frequency is 20 kHz (beyond the range of most humans, but can be heard by dogs)

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–Notice the faster rate at which the ear can detect stimulus changes compared to the eye (ear can detect rates up to 20,000 per second, eye can detect only 50 times per second)

–This needs to be taken into account when converting audio information into digital form-we need to use far more samples per second of information for audio

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Hz (Hertz) is a measure of frequency f (number of cycles (times) a wave repeats itself per second). In our case - how rapidly the audio signal is changing.

If the frequency of a signal is 2Hz, then 2 cycles of the wave are completed per second

Period T, measured in seconds (s) is the time it takes to complete one cycle of the wave

Frequency and period are Inversely Proportional: T=1/f f=1/T

If the frequency of the signal is 2Hz, then the period is 0.5 s (i.e. it takes 0.5 seconds for the wave to complete one cycle)

Amplitude is the magnitude of the signal at a given point in time. For example - volts.

Amplitude relates to volume Louder sounds have greater vertical displacement of sound wave

Period, Frequency & Amplitude

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Period: T=10 ms

0 5 10 t (ms)

Frequency : f=1/(10x10-3)=100 Hz

0 V

1 V

2 V

3 V

-1 V

-2 V

-3 V

Amplitude: A=4 V

-4 V

4 VOne cycle

of the wave

Amplitude: A=-2 V

Amplitude: A=3 V

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Period: T=5 ms

0 5 10 t (ms)

Frequency : f=1/(5x10-3)=200 Hz

0 V

1 V

2 V

3 V

-1 V

-2 V

-3 V

-4 V

4 V

Signal with twice the frequency

Notice the period

is half the value as before

Notice the frequency

is twice the value as before

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http://www.mindspring.com/~scottr/zmusic/

Let’s listen to sine waves with various frequencies:

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Multipliers

Giga (G) 109 1,000,000,000. Mega (M) 106 1,000,000. Kilo (k) 103 1,000. milli (m) 10-3 .001 micro () 10-6 .000001 nano (n) 10-9

.000000001

The following are the common multipliers used for audio characteristics such as period (T) and frequency (f):

For example, KHz = KiloHertz = 1000 Hzms = millisecond = 1/1000 = .001 seconds

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Audio Signal ComponentsConcert “A” on a vibraphone

Sound waves are the sum of simple pure tones

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Frequency composition (spectrum) of a signal

The different frequency components which are added

together to produce a complex waveform are called the

frequency spectrum of that waveform.

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Sound Wave Vs. Frequency Spectrum

Note there is not much frequency content above 1

KHz

Fortunately, we do not need to know the specific frequency content of a signal to digitize it.

We only need to know the highest frequency component the signal might contain (called the bandwidth – a band or range of frequencies that the signal occupies). Why?

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Fourier Transform In the early 19th century, the French mathematician

Fourier proved that all waveforms, whether musical or not, can be constructed out of a sum of pure tones.

The implications of this are that every audio waveform – whether speech, music , or any other sound – can be built out of sinusoids at certain frequencies.

The different frequency components (pure tones) which are added together to produce a complex waveform are called the frequency spectrum of that waveform.

Signals can be converted from time domain (how the wave varies with respect to time) to frequency domain by using Fourier transform (formula).

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Digitizing Audio Signals

In previous lectures, we learned how continuous images are digitized first by dividing the image into a certain number of pixels, then determining the brightness level of each pixel and finally assigning a binary code of certain length (number of bits) to each pixel.

A similar procedure is used to digitize audio signals.

The first step is called “sampling” where the waveform is sampled at certain intervals.

The second step, called “quantization,” involves rounding off the continuous values of the audio samples so they can be represented by a finite number of bits.

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Two step process for digitizing audio

Continuous function of time Infinite amount of information Must choose particular instants of time (samples)

Continuous AudioSignal

Made Discrete In Time

Quantized into a Series of Binary Digits

STEP 1 STEP 2

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Sampling Rate

Sampling Interval (T) Amount of time separating the samples Also called sampling period

Sampling Rate (fs) Number of samples per second Also called sampling frequency

Ts = 1/fs or fs = 1/Ts

Sampling Interval Sampling Rate1 milliseconds 1 kHz = 1000 samples/sec4 milliseconds 250 Hz = 250 samples/sec16 milliseconds 62.5 Hz = 62.5 samples/sec

The sampling rate determines how many values of the signal we choose to retain.

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Digital Telephone Example

In a digital telephone system, the speech signal is sampled 8,000 Hz. What is the sampling period?

Ts = 1/fs

Ts = 1/8000

Ts = .000125 = 1.25 x 10-4 = 125 s

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Nyquist Sampling Frequency

Harry Nyquist, working at Bell Labs developed what has become known as the Nyquist Sampling Theorem:

In order to be ‘perfectly’ represented by its samples, a signal must be sampled at a sampling rate (also called sampling frequency) equal to at least twice its highest frequency component

Or: fs = 2f

Note that fs here is the sampling rate (frequency of sampling), and f is the frequency of the signal

How often do you sample? The sampling rate depends on the signal’s highest frequency (for baseband)

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Sampling Rate Examples

Take Concert A: 440 Hz What would be the minimum sampling rate needed

to accurately capture this signal? fs = 2 x 440 Hz = 880 Hz

Take your telephone used for voice, mostly Highest voice component is: 3000 Hz Minimum sampling rate: fs = 2 x 3000 Hz = 6000 Hz Real telephone digitization is done at 8000 Hz

sampling rate (supporting a 4 kHz bandwidth). Why? We recall that Nyquist determined “equal to at least twice…”

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Undersampling and Oversampling Undersampling

Sampling at an inadequate frequency rate (too low) Aliasing - Aliased into new form (can lead to completely

incorrect signal being reconstructed: has a new identity hiding the original form from us)

Loses information in the original signal

Oversampling Sampling at a rate higher than minimum rate Generation of additional, unnecessary samples - more

values to digitize and process Increases the amount of storage and transmission and therefore, the COST $$

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Effects of Undersampling

Original waveform

Reconstructed waveform

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Reconstructing Audio from Samples

After receiving the signal, it is necessary to reconstruct it in order to hear it.

The signal is reconstructed from its samples.

Exact reconstruction is possible if the sampling rate is sufficiently high.

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A short section of a speech waveform (highest frequency component is 3KHz)

Reconstructed speech waveform with 1 KHz sampling rate (note the resulting waveform does not resemble original waveform)

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Reconstructed speech waveform with 5 KHz sampling rate (the resulting waveform starts resembling the original waveform)

Reconstructed speech waveform with 10 KHz sampling rate (the resulting waveform highly resembles the original waveform)