ĐIỆN tử VIỄN THÔNG audio engineering khotailieu

For correct sampling we must use a sampling rate equal to at least twice the maximum frequency content in the signal.. Signal to Noise Ratio  The ratio of the power of the correct signa

Multimedia Engineering

-Lecture: Basics of Digital Audio

Lecturer: Dr Đỗ Văn Tuấn

Department of Electronics and

Telecommunications

Email: tuandv@epu.edu.vn

1 Digitalization of Sound

2 Quantization and Transmission Audio

Lecture contents

 Sound is a wave phenomenon like light, and involves molecules of air being

compressed and expanded under the action of some physical device

 For example, a speaker in an audio system vibrates back and forth and

produces a longitudinal pressure wave that we perceive as sound Since sound

is a pressure wave, it takes on continuous values, as opposed to digitized ones

 Even though such pressure waves are longitudinal, they still have ordinary

wave properties and behaviors, such as reflection (bouncing), refraction

(change of angle when entering a medium with a different density) and

diffraction (bending around an obstacle)

 If we wish to use a digital version of sound waves we must form digitized

representations of audio information

What is Sound?

 Digitization means conversion to a stream of numbers, and preferably these

numbers should be integers for efficiency

 Figure below shows the 1-dimensional nature of sound: amplitude values

depend on a 1D variable, time (And note that images depend instead on a 2D set of variables, x and y)

Digitization

Figure: An analog signal: continuous measurement of pressure wave

 The graph in the above figure has to be made digital in both time and

amplitude To digitize, the signal must be sampled in each dimension: in time, and in amplitude

 Sampling means measuring the quantity we are interested in, usually at

evenly-spaced intervals.

 The first kind of sampling, using measurements only at evenly spaced time

intervals, is simply called, sampling The rate at which it is performed is called the sampling frequency.

 For audio, typical sampling rates are from 8 kHz (8,000 samples per second) to 48

kHz This range is determined by Nyquist theorem (discussed later).

 Sampling in the amplitude or voltage dimension is called quantization.

 Thus to decide how to digitize audio data we need to answer the following

questions:

 What is the sampling rate?

 How finely is the data to be quantized, and is quantization uniform?

 How is audio data formatted? (file format) Signals can be decomposed into a sum

Sampling and Quantization

Nyquist Theorem

 Signals can be decomposed into a sum of sinusoids

Figure: Building up a complex signal by superposing sinusoids

Nyquist Theorem

 Frequency is an absolute measure, pitch is generally relative – a perceptual

subjective quality of sound

 Pitch and frequency are linked by setting the note A4 exactly 440 Hz

 An octave above that note takes us to another A note An octave

corresponds to doubling the frequency Thus with the middle “A” on a piano (“A4” or “A440”) set to 440 Hz, the next “A” up is at 880 Hz, or one octave above

 Harmonics: any series of musical tones whose frequencies are integral

multiples of the frequency of a fundamental tone

 If we allow non-integer multiples of the base frequency, we allow non-“A”

notes and have a more complex resulting sound

Nyquist Theorem

 The Nyquist theorem states how frequently we must sample in time to be able

to recover the original sound For correct sampling we must use a sampling rate equal to at least twice the maximum frequency content in the signal This rate

is called the Nyquist rate

 Nyquist Theorem: If a signal is band-limited, i.e., there is a lower limit f 1 and

an upper limit f 2 of frequency components in the signal, then the sampling rate

should be at least 2(f 2 − f 1 ).

 Nyquist frequency: half of the Nyquist rate Since it would be impossible to

recover frequencies higher than Nyquist frequency in any event, most systems have an anti-aliasing filter that restricts the frequency content in the input to the sampler to a range at or below Nyquist frequency

Signal to Noise Ratio

 The ratio of the power of the correct signal and the noise is called the signal to

noise ratio (SNR) – a measure of the quality of the signal

 The SNR is usually measured in decibels (dB), where 1 dB is a tenth of a bel

The SNR value, in units of dB, is defined in terms of base-10 logarithms of squared voltages, as follows:

 The power in a signal is proportional to the square of the voltage

 For example, if the signal voltage V Signal is 10 times the noise, then the SNR is

20 log10(10) = 20dB

 In terms of power, if the power from ten violins is ten times that from one

violin playing, then the ratio of power is 10dB, or 1B

Noise

Signal Noise

Signal

V

V V

V

2

10 20 log log



Signal to Quantization Noise Ratio

 Aside from any noise that may have been present in the original analog signal,

there is also an additional error that results from quantization

 If voltages are actually in 0 to 1 but we have only 8 bits in which to store

values, then effectively we force all continuous values of voltage into only

256 different values

 This introduces a round-off error It is not really “noise” Nevertheless it is

called quantization noise (or quantization error)

 The quality of the quantization is characterized by the Signal to Quantization

Noise Ratio (SQNR)

 Quantization noise: the difference between the actual value of the analog

signal, for the particular sampling time, and the nearest quantization interval value

 At most, this error can be as much as half of the interval

 For a quantization accuracy of N bits per sample, the SQNR can be simply

expressed:

NdB

N V

V SNR

N Signal noise

Quan

Signal

02 6 log

20 2

/ 1

2 log

20 log

1

10 _





Signal to Quantization Noise Ratio

Notes:

 We map the maximum signal to 2 N-1 − 1 and the most negative signal to −2 N-1

 Equation above is the Peak signal-to-noise ratio, PSQNR: peak signal and peak

noise

 The dynamic range is the ratio of maximum to minimum absolute values of the

signal: V max /V min The max abs value Vmax gets mapped to 2 N-1 − 1; the min abs

value V min gets mapped to 1 V min is the smallest positive voltage that is not

masked by noise The most negative signal, −V max , is mapped to −2 N-1

 The quantization interval is ∆V = (2V max )/2 N , since there are 2 N intervals The

whole range V max down to (V max − ∆V/2) is mapped to 2 N-1 − 1.

 The maximum noise, in terms of actual voltages, is half the quantization

interval: ∆V/2 = V /2 N

Signal to Quantization Noise Ratio

 6.02N is the worst case

 If the input signal is sinusoidal, the quantization error is statistically

independent, and its magnitude is uniformly distributed between 0 and half of

the interval, then it can be shown that the expression for the SQNR becomes:

SQNR = 6.02N + 1.76(dB)

Audio Filtering

 Prior to sampling and AD conversion, the audio signal is also usually filtered to

remove unwanted frequencies The frequencies kept depend on the application:

 For speech, typically from 50Hz to 10kHz is retained, and other

frequencies are blocked by the use of a band-pass filter that screens out lower and higher frequencies

 An audio music signal will typically contain from about 20Hz up to

20kHz

 At the DA converter end, high frequencies may reappear in the output –

because of sampling and then quantization, smooth input signal is replaced

by a series of step functions containing all possible frequencies

 So at the decoder side, a low-pass filter is used after the DA circuit

Audio Quality vs Data Rate

 The uncompressed data rate increases as more bits are used for quantization

Stereo: double the bandwidth to transmit a digital audio signal

Table: Data rate and bandwidth in sample audio applications

Synthetic Sound

 FM (Frequency Modulation): one approach to generating synthetic sound:

 Wave Table synthesis: A more accurate way of generating sounds from digital

signals Also known, simply, as sampling

 In this technique, the actual digital samples of sounds from real instruments are

stored Since wave tables are stored in memory on the sound card, they can be manipulated by software so that sounds can be combined, edited, and

enhanced

Synthetic Sound

Figure: Frequency Modulation (a): A single frequency (b): Twice the frequency (c): Usually, FM is carried out using a sinusoid argument to a sinusoid (d): A more complex form arises from a carrier frequency, 2πt and a modulating frequency 4πt cosine inside the sinusoid

1 Digitalization of Sound

2 Quantization and Transmission Audio

Lecture contents

Quantization and Transmission Audio

 Coding of Audio: Quantization and transformation of data are collectively

known as coding of the data

 For audio, the μ-law technique for companding (Compressing and

Expanding) audio signals is usually combined with an algorithm that exploits the temporal redundancy present in audio signals

 Differences in signals between the present and a past time can reduce the

size of signal values and also concentrate the histogram of pixel values (differences, now) into a much smaller range

 The result of reducing the variance of values is that lossless compression

methods produce a bit-stream with shorter bit lengths for more likely values

 In general, producing quantized sampled output for audio is called PCM (Pulse

Code Modulation) The differences version is called DPCM (and a crude but efficient variant is called DM) The adaptive version is called ADPCM

Pulse Code Modulation

 The basic techniques for creating digital signals from analog signals are

sampling and quantization

 Quantization consists of selecting breakpoints in magnitude, and then

remapping any value within an interval to one of the representative output

levels

 The set of interval boundaries are called decision boundaries, and the

representative values are called reconstruction levels

 The boundaries for quantizer input intervals that will all be mapped into

the same output level form a coder mapping

 The representative values that are the output values from a quantizer are a

decoder mapping Finally, we may wish to compress the data, by assigning

a bit

Pulse Code Modulation

 Every compression scheme has three stages:

 The input data is transformed to a new representation that is easier or

more efficient to compress

 We may introduce loss of information Quantization is the main lossy step

we use a limited number of reconstruction levels, fewer than in the original signal

 Coding: assign a codeword (thus forming a binary bit-stream) to each

output level or symbol This could be a fixed-length code, or a variable length code such as Huffman coding

 For audio signals, we first consider PCM for digitization This leads to

Lossless Predictive Coding as well as the DPCM scheme; both methods use differential coding As well, we look at the adaptive version, ADPCM, which can provide better compression

PCM in Speech Compression

 Assuming a bandwidth for speech from about 50 Hz to about 10 kHz, the

Nyquist rate would dictate a sampling rate of 20 kHz

 Using uniform quantization without companding, the minimum sample

size we could get away with would likely be about 12 bits

 Hence for mono speech transmission the bit-rate would be 240 kbps (20K

× 12 bits)

 With companding, we can reduce the sample size down to about 8 bits

with the same perceived level of quality, and thus reduce the bit-rate to

160 kbps

 However, the standard approach to telephony in fact assumes that the

highest-frequency audio signal we want to reproduce is only about 4 kHz Therefore the sampling rate is only 8 kHz, and the companded bit-rate thus reduces this to 64 kbps

PCM in Speech Compression

 However, there are two small wrinkles we must also address:

 Since only sounds up to 4 kHz are to be considered, all other frequency

content must be noise Therefore, we should remove this high-frequency content from the analog input signal This is done using a band-limiting filter that blocks out high, as well as very low, frequencies

 A discontinuous signal contains not just frequency components due to the

original signal, but also a theoretically infinite set of higher-frequency components:

 This result is from the theory of Fourier analysis, in signal processing

 These higher frequencies are extraneous

 Therefore the output of the digital-to-analog converter goes to a

low-pass filter that allows only frequencies up to the original maximum to

be retained

PCM in Speech Compression

 The complete scheme for encoding and decoding telephony signals is shown as

a schematic in the figure below As a result of the low-pass filtering, the output becomes smoothed

Figure: PCM signal encoding and decoding

Differential Coding in Audio

 Audio is often stored not in simple PCM but instead in a form that exploits

differences – which are generally smaller numbers, so offer the possibility of using fewer bits to store

 If a time-dependent signal has some consistency over time (“temporal

redundancy”), the difference signal, subtracting the current sample from the previous one, will have a more peaked histogram, with a maximum around

zero

 For example, as an extreme case the histogram for a linear ramp signal that has

constant slope is flat, whereas the histogram for the derivative of the signal

(i.e., the differences, from sampling point to sampling point) consists of a spike

at the slope value

 So if we then go on to assign bit-string codewords to differences, we can

assign short codes to prevalent values and long codewords to rarely occurring ones

Lossless Predictive Coding

 Predictive coding: simply means transmitting differences – predict the next

sample as being equal to the current sample; send not the sample itself but the difference between previous and next

 Predictive coding consists of finding differences, and transmitting these using

a PCM system

 Note that differences of integers will be integers Denote the integer input

signal as the set of values Then we predict values as simply the previous value, and define the error en as the difference between the actual and the predicted signal:

 But it is often the case that some function of a few of the previous values

provides a better prediction Typically, a linear predictor

f

 The idea of forming differences is to make the histogram of sample values

more peaked

 For example, the first figure plots 1 second of sampled speech at 8 kHz,

with magnitude resolution of 8 bits per sample

 A histogram of these values is actually centered around zero

 The last figure shows the histogram for corresponding speech signal

differences: difference values are much more clustered around zero than are sample values themselves

 As a result, a method that assigns short code-words to frequently

occurring symbols will assign a short code to zero and do rather well: such

a coding scheme will much more efficiently code sample differences than samples themselves

Lossless Predictive Coding