What is a digital signal?

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Digital Signal Processing
A host of technologies touching many interdisciplinary areas
• What is a signal?
a.
b.
c.
d.
An agreed upon message that triggers action or carries information
A message that conveys notice or warning
An electronic message conveyed by telephone, radio, radar, or television
A measurable physical quantity (e.g. voltage, current, magnetic field)
• What is a digital signal? An array of measurements
• Examples of Signal Processing? Stock market prices, streaming of
video, data mining, natural language processing, data compression, image
processing, earthquake prediction, medical diagnosis, etc.
• Interdisciplinary aspects: Physics, Engineering, Psychology, Mathematics,
Computer Science, Cognition, Biology, etc.
• Processing algorithms: breaks down to twiddling single and multi
dimension arrays; often algorithms result in small amounts of code, which is
heavy in mathematics/statistics
• Difficulties: Signals are often imprecise and have many redundancies
Digital Signal Processing (DSP)
• Radar and Sonar
– What signal comes back?
• Image Processing
– Transmit & reconstruct
• Linguistics
– Recognize & Synthesis
• Algorithms
– Compress and encrypt,
noise removal, filter
• Music
• Military:
– Secure channels, laser bombs
• Exploration:
– oil, minerals, ocean mapping
• Climate:
– Earthquake, weather patterns
• Grand Challenges
– Real world simulations
• Medical
– Medical Resonance Imaging
– Synthesis and edit
Applications: Cell Phones, voice mail, phone support,
web translation, and many more.
Signal Redundancy
• Continuous signal (virtually infinite)
• Sampled representations of speech
– Mac: 44,100 2-byte samples per second (705kbps)
– PC: 16,000 2-byte samples per second (256kbps)
– Telephone: 4k 1-byte sample per second (32kbps)
– CELP Audio Compression Algorithm: 8kbps
– Research: 4kbps, 2.4 kbps
– Military applications: 600 bps
– Human brain: 50 bps
Definition: Sampling Rate is the number of measurements per second
Noise
• Definition: Random electrical or acoustic activity that
obscures communication
• Noise impact on speech
–
–
–
–
Noise degrades the speech recognition effectiveness
Consider construction noise outside the window
How much noise makes normal speech unintelligible?
What do human beings do to compensate?
• Automated Solution
– Apply appropriate filters to eliminate noise from signals
– Effective filters must not destroy essential signal data
Natural Language Processing
The Noisy Channel
Computational Linguistics
1. Replace the vocal articulators with a synthesizer
2. Replace the ear with a receiver
2. Replace the brain with the computer
Could a computer process this?
I cdnuolt blveiee that I cluod aulaclty uesdnatnrd what I was
rdgnieg.
The phaonmneal pweor of the hmuan mnid Aoccdrnig to
rscheearch at Cmabridgde Uinervtisy, it deosn't mttaer in
what oredr the ltteers in a word are, the olny iprmoatnt
tihng is that the frist and lsat ltteer be in the rghit pclae.
The rset can be a taotl mses and you can still raed it wouthit
a problem.
This is bcuseae the huamn mnid deos not raed ervey lteter
by istlef, but the word as a wlohe.
Amzanig huh?
Yaeh and I awlyas thought slpeling was ipmorantt!
Discrete versus Continuous
• Continuous (analog): A signal, typically in found in nature,
whose infinite values are represented by a mathematical
function. That is given any point in time, if we know the initial
conditions, we can compute a value at that point.
• Discrete (digital): A sequence of measurements, each one
separated by a fixed time interval.
• Digital processing of continuous signals
– Computers deal with digital signals, because memory, though
large, has a finite size.
– Input devices measure signal strength thousands of times a
second (e.g. audio: 44,100 is common). These values translate
into a huge arrays in computer memory.
Signal Dimensionality
Note: Even higher dimension signals can be reduced to a single dimension
• Single Dimension (audio)
• Three Dimensions (video)
• Double Dimension (images)
• Higher Dimension (Climate
Change Simulations)
Signal Power (Energy)
Energy is the square of the amplitude
• Decibels: ratio between power of two
signals. Dimensionless, like percent.
• SPL (Sound Pressure Level) is the ratio of
a sound to the threshold of human hearing
Note: doubling the power increases decibels
by three. Increase by 10 decibels increases
power by 10.
Note: 10 log(P1/P2) = 20 log(A1/A2)
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Sound
dB
TOH
0
Whisper
10
Quiet Room
20
Office
50
Normal
conversation
60
Busy street
70
Heavy truck traffic
90
Power tools
110
Pain threshold
120
Sonic boom
140
Permanent
damage
150
Jet engine
160
Cannon muzzle
220
Working with Signals
• Special purpose languages: MatLab, Octave
– Matlab is very expensive, but available on campus
– Octave, is largely compatible (but not entirely), and is free
• Free signal visualization programs: GraphCalc
• Sound Editing tools: ACORNS Sound Editor, Audacity.
Sound Editor source is available to the class
• Java based resources: Java Sound, Tritonus
GraphCalc (Freeware )
Good for creating and visualizing signals
Communication of Signals
• Pulse code Modulation (PCM)
– An array of measured values
– Audio wav files: PCM with a header, defining big versus small
endian, sample rate, bytes per sample, number of channels
• Compression techniques to reduce bandwidth
– Transmit differences between adjacent values
– Don’t transmit runs of the same value
– Algorithms (e.g. Linear prediction) to predict next values
from previous, and transmit the prediction errors
– Coding techniques to transmit infrequent values using more
bytes than those that are common
Processing first step: translate compressed values to PCM
Understanding Sine Waves
• Sine is the ratio of the height to the hypotenuse
• Many phenomena in nature occur in sine wave patterns
Basic Terminology – periodic signals
• Amplitude — The distance from zero to the maximum height
• Period — The time it takes for a sine wave to complete one cycle
• Frequency (Hz) — The repetitions or cycles per second (1/period)
Rope Impulse Waves
• A hand jerk is an impulse starting the wave to travel.
– Speed of hand jerk determines frequency
– Rope stiffness determines wave velocity
– Reverse waves starts at the fixed end, which can resonate if forward
waves correlate, or cancel if they don’t.
– Eventually a steady state is reached and we perceive no motion
Traffic Waves
Complex Wave Patterns
• Sound waves occupying the
same space combine to form a
new wave of a different
shape.
• Harmonically related waves
add together and can create
any complex wave pattern.
• Harmonically related waves
have frequencies that are
multiples of a basic frequency.
Note: All frequency waves do not have to start at zero,
they can be “out of phase”. The amount of shift in
degrees is called their phase angle.
Complex Wave Examples
Time vs. Frequency Domain
Time Domain: A composite wave summing different frequencies
Frequency Domain: Split time domain into component frequencies
Historical Note
• Jean Baptist Joseph Fourier (1768-1830)
– Paper submitted: Academy of Science in Paris, 1807
– Claim: All signals decompose into a sum of sine waves
• The review committee
– Laplace (1749-1827) voted to accept
– Lagrange (1736-1813) voted to reject. The claim did not
account for waves with sharp corners
– The paper never got published
• Results
– Fourier was correct if we use an infinite series of waves
– Lagrange was correct if we use a finite series of waves
– Fourier analysis is the foundation for Digital Signal
Processing (DSP)
Time and Frequency
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Non Periodic or Quasi-periodic
• Many signals change their characteristics over time
• Fourier showed that within smalls time windows, we can
analyze signal chunks as if they were periodic
• Processing involves
1. Chopping signals into overlapping “frames” of some millisecond
width
2. “Window” the frames, which means apply algorithms to
smooth the abrupt edges
3. Perform “feature” extraction algorithms on windowed frames to
obtain set of numbers that functions like a frame’s fingerprint
4. Use the feature sets to perform additional processing
Quasi-periodic: Signals, that are somewhat periodic, but whose
features slowly change over time
Sample Sound Signal (Sound Editor)
Download and install from ACORNS web-site
Top: “this is a demo”
Bottom: “A goat …. A coat”
Time Domain: x-axis: time, y-axis: amplitude measurement
Frequency Domain
x-axis: time, y-axis: frequency, darkness = amplitude
Narrow Band Spectrogram (horizontal lines – emphasizes frequency bins)
Wide Band Spectrogram (Vertical lines – alignment of pitch harmonics)
Signal Power or Energy
Power is the square of the amplitude, often measured in decibels
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Temporal Features
• Advantages
– Minimal processing
– Easy to understand
• Examples
–
–
–
–
–
Zero-crossing rate
Autocorrelation and autocorrelation peaks
Pitch periods
Loudness contour
Maximum and minimum distance between audio
positive and negative amplitude
– Degree of voice in sounds (voicing quality)
Variance between Speakers
Signal Filter
Modify the signal input in some way to produce a desired output
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Filter Example
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Moving Average Filter
Useful Examples:
• A Crude low pass filter
• Smoothing a speech pitch contour
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Signal processing Operations
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Linear Systems
• If x1[n] ---> system ---> y1[n] and
x2[n] ---> system ---> y2[n] where system
manipulates the input values in some way
• A system is linear must be
– Homogeneous (output is the sum of its parts)
c1*x1[n] + c2*x2[n] ---> system ---> c1*y1[n] + c2*y2[n]
– Time invariant (delay of input delays the output)
x’1[n] = x[n+k] ---> system ---> y’1[n] = y[n+k]
Linear Systems
A Linear System is one that is homogenous and additive
Homogeneity
• Multiply input to multiply
the output
• If (x(n)  y(n)
• Then c * x(n) -> c*y(n)
Additivity
• Summing input signals sums
the output signals
• If x1(n)y1(n) & x2(n)y2(n)
• Then x1(n)+x2(n)y1(n)+y2(n)
Time Invariance
Not required to be linear, but highly desirable
• Shift the input signal to identically shift the output
• Assume: x(n)y(n)
• Then: x(n+∆) y(n+∆)
Time Invariant Example
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Sinusoidal Fidelity
• Feed a sine wave to the input
• Linear Systems
– Output will also be a sine wave
– Output phase can shift
– Output amplitude can be different
• Non-linear systems
– Not likely to demonstrate sinusoidal fidelity
– It is possible, however, for a non-linear system
to generate sine waves that don’t correlate to
the input in a linear fashion
Linear System Example
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Note: Linear systems can be connected in series and the result is still a
linear system. Flipping the order of components does not affect the result
because linear systems are commutative.
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Note: Sending an impulse to an unknown system can help determine
its operational characteristics
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Non Linear System Example
Useful Examples
• Median of five is
useful in computing
a smooth pitch
contour
• Speech signal are
mostly linear, but
have some nonlinear components
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
How to Handle non-Linear Systems?
They must be linear for known ways of analysis
Approaches
• Ignore the non-linearity if it is small enough
– Example: low levels of noise
• Convert to small amplitudes.
– Many systems appear linear with small amplitudes
• Apply a linearizing transform
– Assume a[n] = b[n] * c[n]
– Take logarithms (log(a[n] = log(b[n]) + log(c[n])
Synthesis and Decomposition
Note: Algorithms often decompose a signal into pieces,
perform some calculations, and then reconstruct
Note: We can always reconstruct the
by adding up the decomposed pieces
Decomposition: break signal
into two or more additive components
Synthesis: Combining signals
using scaling and addition
DSP with Synthesis and
Decomposition
Analyze in pieces to get result
Numeric Example:
2041*4 = 2000*4+40*4+1*4
Decompose, Feed into linear
Systems, and Synthesize Output
Even/Odd
Decomposition
• Assumptions
– Even number of total points (N)
– End sample attached to beginning
– Decompose into even and odd signal
• Even Symmetry
– xE[n]= (x[n] + [N-n])/2
– Point 0 and N/2 = signal value
• Odd Symmetry
– xO[N]= (x[n]-x[N-n])/2
– Point 0 and N/2 = 0
Interlaced
Decomposition
• Decompose into
even and odd
signals
• Even signal contains
zeroes in the odd
indices
• Odd signal contains
zeroes in the even
indices
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