Title: Audio Sampling
1Audio Sampling
- Read Chapter 11 in text book!
- Chapter 3 of optional text has excellent material
that complements the required text - Chapter 10 of text provides some additional
information you may find useful (In other words,
read the chapter) - We will cover in some detail exactly how
continuously variable (analog) signals become
digital (discrete) binary code
2Background
- Audio signal (sound) reflects the motion of air
particles in space - Microphone converts this motion of air particles
into an electrical signal - Audio signals have frequency components that are
complex - In other words, most audio signals are made up of
many different frequencies, combining to make the
sound we recognize (I just saved you from Fourier)
3Audio Signal
- Audio signals have certain characteristics that
are well known - Human voice varies from about 100 Hz to 4000 Hz
- Piano Concert A above Middle C is 440 Hz
- Hertz (Hz) means cycles per second
- Sound varies in a manner similar to a Sine Wave
- Concert A varies 440 times per second
- My voice sounds different than a pianoWhy?
4Characteristics of Signals
5Signal with Twice of Frequency
6Audio Signal Components
Concert A on a vibraphone
The different frequency components which are
added together to produce a complex waveform are
called the frequency spectrum of that waveform.
Every sound wave is the sum of simple pure tones.
7Voice Signal Components
8Highest Frequency Requirement
- 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 signal
in a sample.
9More information
- Bandwidth
- Is often represented by the highest frequency
signal in a sample. But in common usage is used
to describe the data rate a channel supports - Can be thought of as the range of frequencies
some transmission medium (air, water, cable,
fiber optics) can support - Concert A has a bandwidth of 440 Hz
- A human voice has a varying bandwidthbut we said
that we well know that the average humans
highest frequency component is 3000 Hz
(3kHz)What is the bandwidth?
10Two Step Process To Digitizing Audio
- Continuous function of time
- Infinite amount of information
- Must choose particular instants of time
STEP 1
STEP 2
Continuous Audio Signal
Quantized into a Series of Binary Digits
Made Discrete In Time
11Periods Signal and Sampling
- Take the 440 Hz Concert Ahow long is one cycle?
- Ts 1/fs ? Ts 1/440 Hz 0.00227 sec
- Also fs 1/Ts ? fs 1/ 0.00227 sec 440 Hz
- In English The Time period of one cycle is
equal to the inverse of the frequency of the
signal in Hz - Problem now is that the Concert A is a
continuously variable signal, and we want to
digitize it - How often should we sample the signal?
12Nyquist
- Harry Nyquist, working at Bell Labs (youve heard
that before, havent you?) 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 equal to at least twice its highest
frequency component - fs 2f
- Also represented as
- fs 2B Hz where B is bandlimited to a
highest - frequency
- Note that fs here is frequency of sampling, not
the frequency of the sample
13For Example
- Take Concert A 440 Hz
- What would be the minimum sampling rate needed to
accurately capture this signal? - Plug and chug 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 - Bell Labs built in some slop ? real telephone
digitization is done at 8000 Hz sampling rate
(supporting a 4 kHz bandwidth). Why? Remember
that Nyquist said equal to at least twice
14Example Speech Waveform
15Nyquists Revenge
- If you oversample (exceed the Nyquist rate), we
create more bits to store and to transmit than is
necessary to accurately send the signal to the
distant end - But, moderate oversampling can work to our
advantage, making the system more robust and
easier to implement ? Thats why Bell Labs
decided to sample at 8 kHz vice 6 kHz
16Nyquists Revenge, Cont
- If you undersample (sample at less than the
Nyquist rate), you run the danger of aliasing - Aliasing is when the signal you decode is not the
signal you encoded - Aliasing then, is bad!
17UndersamplingA Very Bad Thing
Thanks to http//euphoria.org/home/help/nyquist.h
tml
18Quantization
- Read Chapters 12 and 14 in text book!
- Chapter 3 of optional text has excellent material
that complements the required text - Chapter 10 of text provides some additional
information you may find useful (In other words,
read the chapter) - We will cover in some detail exactly how
continuously variable (analog) signals become
digital (discrete) binary code
19Quantization
- Audio Signals Continuous in time and amplitude
- Digitization of Audio Signals Must be made
discrete in time and amplitude - Weve learned how to make Continuous signals
discrete in Time Sampling (Dont forget about
Nyquist!) - To make Continuous signals discrete in Amplitude,
we Quantize
Step 1 Sampling
Step 2 Quantization
The two step digitization process
20What is Quantization?
- Quantization establishes a range of values of the
continuous function that we say can be
represented by a particular binary code - Looking at it another way We round things off
- For example We can say that the continuous
temperature range 40.0 to 40.1 degrees F can be
represented by one binary code (lets just say
0110010000)--The temperature obviously can vary
over 40.0 to 40.1 (temp might actually be
40.00987 degrees), but we say that measuring to
within a tenth of a degree is OK
21Picking the Code
- We have a trade-off to make (recurring theme)
- Number of bits we want to use versus precision of
digitization (What does that mean?) - Go back to when you were Thermometer Engineers
We want to measure -40º F to 140º F (over 180º F) - We decided to implement an 8 bit code at first
- If we use all the bits, what degree of precision
do we achieve?
22Precision of Quantization
- 8 bit code can represent 256 codewords
- Our thermometer can measure to
- (Range of Measurement)/Number of Codewords
- Or 180º F/256 Codewords 0.703º F per codeword
- The text on p. 174 uses an example of sampling a
voltage that ranges from -10V to 10V - The range measured is over 20V, encoded with 4
bits - Or 20V/16 Codewords 1.25V per codeword
- For the text example, lets look at the
Quantization Codetable
23A Time Quantized Audio Waveform
24Problems with Quantization
- Some information is lost
- Error is introduced (Noise is introduced)
- Codetable on p. 174 uses the Range Center as the
value of quantization to minimize both effects
(Noise and Error)
25Problems with Quantization Cont.
- Difference between the actual value of signal and
the quantized value is the Quantization Error - In book example Say actual voltage value when
sampled (a point in time) is 8.5V - Error is 8.5V - 8.125V 0.375V
- Another example Actual voltage is 7.75V
- Error is 7.75V - 8.125V -0.375V
26Quantizing and Re-Constructing the Signal
- Analog-to-Digital Converter (ADC) provides the
sampled and quantized binary code - Digital-to-Analog Converter (DAC) converts the
quantized binary code back into an approximation
of the analog signal by clocking the code to the
same sample rate as the ADC conversion - Quantize and Reconstruct the analog signal
Example on the next couple of slides
27Quantizing
28Reconstructing
29Another Example
30Data Rate, Decibels, Signal-to-Noise Ratio,
Channel Capacity (Shannons Law)
- You really should read the sections of Chapter 8
of the optional textbook - Nyquist is only part of the equation when you are
developing a digital transmission system - Nyquist tells you how often you should sample,
but not how much information a channel (a medium
through which data moves) can support
31Little Background on Data Rate
- There is a difference between signal rate and
data rate - The unit of measure for signal rate is baud
- The unit of measure for data rate is bits per
second - The difference? A signal rate is equal to the
number of signal events in a period of time. - One bit per signal event--baud rate and bit rate
are the same! - More than one bit per signal event--bit rate
exceeds baud rate! - Dont believe me? Lets go to the next two charts
32One Bit per Signal Event
Amplitude (V)
1 -- V1
Why arent the vertical lines vertical?
How many signal events here and here?
0 -- V0
time
One signal event--one bit sent per signal event
33Signal Events with More than One Bit
Amplitude (V)
111 -- V7
110 -- V6
101 -- V5
100 -- V4
011 -- V3
010 -- V2
001 -- V1
000 -- V0
time
One signal event--three bits sent per signal event
34Channel Noise and Decoding
Little noise decoding with M8 More noise
results in error The same amount of noise but
decoding with M2
35Relating Baud to Data Rate
- The signal transmission rate (baud rate) can be
related to the data rate - We have to know the baud rate (signals per
second) - We have to know how many bits are being sent per
signal event, then know how many symbols this
number of bits can take on (optional text refers
to this as M-ary transmission--more in the
modulation lecture) - We have to know how to take log2 of a value
- log2 x log10 x/log10 2
- We have to know D R log2 M bps
- Where D is data rate, R is baud rate and M is
number of symbols per signal event
36For Example
- A system signals at 1 baud, 1 bit per signal
event (2 symbols per event) - D R log2 M bps --gt D 1 log2 2 bps 1
bps - A system signals at 75 baud, 8 bits per signal
event (256 symbols per event) - D R log2 M bps --gt D 75 log2 256 bps
600 bps - A system signals at 4000 baud, 8 bits per signal
event (256 symbols per event) - D R log2 M bps --gt D 4000 log2 256 bps
32000 bps
37Decibels
- Whats a decibel?
- In electronics and communications, the decibel
(abbreviated as dB) is a logarithmic expression
of the ratio between two signal power, voltage,
or current levels. In acoustics, the decibel is
used as an absolute indicator of sound power per
unit area. A decibel is one-tenth of a Bel, a
seldom-used unit named for Alexander Graham Bell,
inventor of the telephone (What is) - Power (unit of measurement watts (W))
- dB 10 log P2/P1
- Voltage or Amperage
- dB 20 log V2/V1
- dB 20 log A2/A1
Important! Note that dB are log base 10, not base
2!
38For Example
- You want to express in dB the power relationship
between your stereo and the TV (roommate fight) - Stereo has 200W of power, TV has 50W of power
- dBW 10 log 200/50 6.02 dBW
- Stereo wins this battle
- Note the other way-- Roomies TV has 50W, Stereo
200W - dBW 10 log 50/200 -6.02 dBW
- Roomies TV loses the battle
39Signal-to-Noise Ratio
- S/N is normally measured in dB, as a relationship
between the signal you want versus the noise that
you dont but is in the medium (to include your
radioheat causes noiseever notice that your
radio warms up when it is on?) - It can be thought of as a fractional relationship
(that is, before you take the log of it) - 1000W of power v. 20W of noise is either
- 50 (unitless!)
- or about 17 dBW gt 10 log 1000/20 16.9897
dBW
40Shannons Law(Shannons Limit for Information
Capacity)
- Claude Shannon at Bell Labs (where else?) figured
out how much information a channel could
theoretically carry - I B log2 (1 S/N)
- Where I is Information Capacity in bps
- B is bandwidth in Hz
- S/N is Signal-to-Noise ratio (unitlessdont make
into dB)
Note we are back to log base 2!
41Shannon in Action
- You have a telephone system with a 3 kHz
bandwidth and a S/N ratio of 1000 (very good
indeed!) - I B log2 (1 S/N) --gt I 3000 Hz log2 (1
1000) I 3000 Hz log2 1001 --gt 29902 bps
(29.9 kbps) - You have a telephone system with a 3 kHz
bandwidth and a S/N ratio of 10 (very bad!) - I B log2 (1 S/N) --gt I 3000 log2 (1 10)
I 3000 log2 (11) --gt 10378 bps
(10.4 kbps)
42Shannons Law Caveats
- In order to achieve the theoretical Shannon
limit, the transmission system is not binary, it
is some form of M-ary (i.e. more than one bit is
sent per signal event) - You see that in order to increase capacity, you
must either increase the bandwidth of the channel
or increase the S/N ratio (make the signal
stronger or reduce the noise on the channel)
43Modulation
- Re-read Section 14.7 of the text
- Modulation is process of impressing a
low-frequency information signal onto a higher
frequency carrier signal - Modulation is done to bring information signals
up to the Radio Frequency (or higher) signal - Some systems even have two stage modulation,
where the information is brought up to an
Intermediate Frequency (IF), and then increased
to the transmission frequency
44Some Definitions
- Baseband Signal is a term used to describe the
unmodulated signalor in other words, the
information signal - Carrier Signal is what the information signal is
combined with to form the new modulated
signalthe frequency of the carrier is described
as the center frequency of the signal
45Types of Analog or Traditional Modulation
- Amplitude Modulation (AM)
- Information signal is added and subtracted to and
from a carrier signal - Frequency Modulation (FM)
- Information signal varies a constant amplitude
carrier signals frequency directly in proportion
to the informations frequency - Phase Modulation (PM)
- Information signal varies a constant amplitude
carrier signals phase directly in proportion to
the informations frequency - Both FM and PM are forms of Angle Modulation
46Amplitude Modulation
Carrier
Baseband
Modulated
47AM Continued
- Note that my diagram is highly out of scale
- Carrier Signal is centered on one frequency, and
has a bandwidth (like 1000 kHz, 5 kHz) - Baseband Signal varies from 0 -- some value Hz
(human voice 0 -- 4 kHz) - AM naturally has two sidebands, but only one
sideband is needed to get back all the
information - Two bands? 1000 kHz 3 kHz 997 kHz and 1003
kHz - Figure 14.14 shows this in another way
48AM Modulation and AM Channels
49Frequency Modulation
50FM Continued
- Same scale issues on diagram
- FM is constant amplitude--another way of thinking
about it FM always has the same bandwidth - FM signal varies around the maximum and minimum
deviations of the baseband signal - Modulated signal is more compressed around the
maximum deviationmore spread out around the
minimum deviation - In other words, the variance of the frequency of
the constant amplitude carrier is directly
proportional to the amplitude of the modulating
signal at a rate equal to the frequency of the
modulating signal
51FM Modulation and FM Channels
52Phase Modulation
- It differs from FM in the time domain, given the
same baseband and carrier signals, but is
otherwise identical - Will see it next
53Summary
Lets compare them
54Frequency Shift Key
- Similar to FM
- A low frequency for a time means 0
- A high frequency for a time means 1
- Derived from TTY days Shift and Mark
0 1 0 1
0 0 1 1