digital signal processingcsce.uark.edu/~ahnelson/csce4114/lectures/lecture13.pdf · adc { analog to...
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Digital Signal Processing
Alexander Nelson
October 14, 2019
University of Arkansas - Department of Computer Science and Computer Engineering
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Digital Signal Processing
Digital Signal Processing – signal processing in the discrete domain
Signal Processing – “analysis, synthesis, and modification of
signals”
Signals – “functions conveying information about behavior or
attributes of some phenomenon”
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Signals
Signals – “functions conveying information about behavior or
attributes of some phenomenon”
Real world signals are typically analog
Often sinusoidal in nature
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Information
Information – “Any entity or form that provides the answer to a
question or resolves uncertainty”
Signals provide information about a phenomenon
Information resolves uncertainty about a question or phenomenon
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Information: Example
Example:
Question – What is the temperature of an object?
Temperature – physical quality of a material
Signal – Analog quantity of temperature over time
Thermometer – Instrument that measures the physical quantity
Digital Signal – Discrete value of the signal measured by the
thermometer
Information – How much uncertainty the signal resolves
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Entropy – Information Theory
Entropy – Average amount of information produced by stochastic
source of data
Coin toss - 1 bit of entropy per toss
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Entropy Example
Assume I have a properly shuffled card deck
What is the uncertainty?
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Entropy Example
Assume I have a properly shuffled card deck
What is the uncertainty?
What is the probability of drawing any particular card? = 1/52
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Entropy Example
Assume I have a properly shuffled card deck
What is the uncertainty?
What is the probability of drawing any particular card? = 1/52
Bits of information = log2(52) = 5.7 bits of information
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Entropy Example
How many bits if you knew it wasn’t a heart?
How many bits if you knew it was a spade?
How many bits if you knew it was the 3 of clubs?
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Digital Signal Processing
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Sensors
Sensor – Component that measures physical phenomena
Usually provides an Analog signal – Need to convert to digital
Raw analog signals carry information
Signal processing enables understanding of that information
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Sensor Limits
Limitations:
• Range – How wide the physical quantity can be measured
• Resolution – How specific can the measurement be
• Accuracy/Precision – How close is measured value to actual
value
• Raw – Typically measuring voltage or other physical quantity
that relates to the actual value – Requires processing
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Example
Range – 4-34pF
Sensitivity – 4pf/10000
count = 0.4fF
Non-linear – Above 25pf,
non-linear response
Raw – Capacitance must
be related to physical
quantity
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Example
Capacitance to position
Fabric sensors detect distance to
person
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Actuator
Actuator – Component with in input that converts input to energy
e.g.
• Servo – Kinetic Energy
• LED – Light Energy
• Thermal – Heating elements
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Example – Servo
Servo – Converts PWM signal to angle
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Transform
Transform – Takes input, modifies, & generates output
Examples:
Smart streetlight – Samples light sensor, turns on streetlight
gradually at sunset
Stereo – Samples CD input and volume dial, plays sound on
speakers
Conceptually:
Input→Transform→Output
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ADC – Analog to Digital Converter
Mixed-Signal Transform – Converts analog signal to digital signal
DAC is also a mixed-signal transform – converts digital to analog
ADC has the following characteristics:
• Input Range – Voltage of the high and low values
• Output Range/Quantization – Number of bits that the value
can map to
• Sampling Rate – Frequency of measuring the value
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ADC – Amplification
Many signals don’t have enough peak-to-peak variance to
recognize with high sensitivity
Signal under-loading – ADC isn’t sensitive enough
Amplifier – Transform – Multiplies signal by some value (called
Gain)
Gain – Multiple that signal is multiplied by
Gain > 1 – Amplifier, Gain <1 – attenuator, Gain = 1 – buffer
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ADC – Amplification
Goal of gain – Obtain peak-to-peak value of signal near the input
range of ADC
Signal overloading – Too much gain; signal outside range of ADC
Clipping – Loss of signal values beyond input range
Signal underloading – Not enough gain; ADC not sensitive
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Overloading
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Bias
Bias – Signal average is not centered at center of ADC value
i.e. For ADC with range [-1,1], if x̂ != 0, biased in some direction
For periodic signals (e.g. Alternating Current) removing bias often
referred to as removing the DC (Direct Current)
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Bias/DC
Recall Lab 7 with the FFT of the
siren
The DC is the first component of
the FFT
Can be completely subtracted
from the signal with no effect on
the periodic signals
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Recap
Signal is:
• Analog
• Has bias
• Needs amplification to range of ADC
• Needs to be sampled by ADC
These processes are called “signal conditioning”
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Signal Fidelity
Fidelity – Correspondence of output signal to input signal
(“accuracy”)
Affected by two parameters:
• Sampling Rate – Affects loss of value in time
• Resolution/Quantization – Affects loss of value in space
These two are related – Faster ADCs often have fewer bits
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Dynamic Range
Dynamic Range – Measure of the precision of a device w/ respect
to range of values it can convert
Difference b/w dynamic range & range?
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Dynamic Range
Dynamic Range – Measure of the precision of a device w/ respect
to range of values it can convert
Difference b/w dynamic range & range?
Range = peak-to-peak gap
Dynamic range = Ratio of max range to minimum precision of the
converter
All (static range) N-Bit ADCs have the same dynamic range
Often expressed in decibel logarithmic scale
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Dynamic Range
range = Vmax − Vmin (1)
gap =range
2N(2)
dynamic range =range
gap(3)
=Vmax − Vmin(
Vmax−Vmin
2N
) (4)
=Vmax − Vmin
Vmax − Vmin× 2N (5)
= 2N (6)
Dynamic Range directly related to number of bits
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Dynamic Range
dynamic range (dB) = 10× log
(range2
gap2
)(7)
= 10× log
(range
gap
)2
(8)
= 20× log
(range
gap
)(9)
= 20× log(2)N (10)
= N × 20× log(2) (11)
= 6.02× N (12)
e.g. 12-bit ADC has dynamic range of 6.02× 12 = 72.2dB
Human ear has approx. 130dB dynamic range
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Sampling Rate
Sampling Rate should be responsive to application
Sampling sound at 2KHz results in just 2 samples in 1 ms
Clearly underlying signal cannot be reconstructed
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Sampling Rate
Nyquist Rate – 2× f of highest frequency signal
Nyquist rate guarantees signal can be reconstructed from samples
Hearing Range of Humans ≈ 20Hz-20kHz
Need to sample at least 40kHz for Nyquist rate
MP3 sampling rate = 44.1kHz
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Noise & Aliasing
Noise – Unwanted signal; e.g. Microwave oven vs. WiFi/Bluetooth
Noise Reduction – Techniques to remove noise from signal path
Shielding – Electrical considerations to reduce noise
Signal-to-noise ratio – Ratio of desired signal to noise
Aliasing – Presence of unexpected signal in the signal path
Source of noise if unhandled
Example: Car wheel in commercials
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Aliasing Noise
If we are only trying to reconstruct lower frequency signals, can we
just ignore higher frequencies?
High frequency signals subsampled can alias onto our existing
signal, creating noise
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Removing Aliasing Noise
Two approaches:
1. Sample fast enough to digitally eliminate noise – Overampling
2. Filter higher frequency signal
Oversampling is expensive in terms of storage & processing
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Low Pass Filter
Filter – “pass through a device to remove unwanted material”
Signal Filter – Remove unwanted signal/noise
Low Pass Filter – Filter that lets low frequency signals through
sometimes called anti-aliasing filter
Center Frequency – Frequency that all signals below should go
through, all above should stop
Ideal filter 34
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Low-Pass Filter
Realistic Low-Pass Filter
Unfortunately, real filters don’t respond perfectly
Some noise will get into the system
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Complete Input Path
We now have the trappings of a full input DSP Path
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Playback Path
What about the other direction?
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Playback Path
What about the other direction?
Playback path – reconstruct the original signal
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Playback Path
Speaker – Actuator for a sound system
Amplifier/Filter swap places – Don’t amplify sound then try to
remove
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Digital to Analog Converter (DAC)
Digital to Analog Converter has:
• Input Range – Number of bits
• Output Range – Real quality (e.g. Volts)
• Sampling Rate – Rate at which DAC consumes input,
produces output
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Complete DSP System
What about modifications?
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