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Session 2 : Review On Signals
Session delivered by:
Chandan N.
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Session Objectives To understand basics operation on signals To understand the Time-Domain Characterization of LTI
system To understand the effects of under sampling and over
sampling To understand the concept of convloution To review on Time domain and Frequency domain signals
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Session Topics Types of Signals Discrete time Systems Sampling Signal processing
Aliasing
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Basic Sequences
Unit sample sequence -
Unit step sequence -
==
0,00,1][
nnn
1
4 3 2 1 0 1 2 3 4 5 6 n
0 or shift to the left by n sampling periods if
n < 0 to form 3) Form the product 4) Sum all samples of v[k ] to develop the n-th
sample of y[n] of the convolution sum
][ k h
][ k nh ][][][ k nhk xk v =
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Convolution Sum
Schematic Representation -
The computation of an output sample using the convolution sum issimply a sum of products
Involves fairly simple operations such as additions, multiplications,and delays
n ][ k nh ][ k h
][k x
][k v][n y
k
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Ti D i Ch t i ti f LTI
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Time-Domain Characterization of LTIDiscrete-Time System
In practice, if either the input or the impulseresponse is of finite length, the convolution sumcan be used to compute the output sample as itinvolves a finite sum of products
If both the input sequence and the impulseresponse sequence are of finite length, the outputsequence is also of finite length
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Time-Domain Characterization of LTIDiscrete-Time System
If both the input sequence and the impulseresponse sequence are of infinite length,convolution sum cannot be used to compute theoutput
For systems characterized by an infinite impulseresponse sequence, an alternate time-domaindescription involving a finite sum of products will
be considered
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Convolution Example
Example - Develop the sequence y[n]generated by the convolution of thesequences x[n] and h[n] shown below
0 1 23
12
1
n01
23
4
2
1
3
1
n
x[n] h[n]
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Convolution Example
The sequence { y[n]} generated by theconvolution sum is shown below
2
4
1 11
3
5
3
2 3 4 5 6
0 1
2 1
7
8 9n
y[n]
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Simple Interconnection Schemes
Two simple interconnection schemes are: Cascade Connection Parallel Connection
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Cascade Connection
Impulse response h[n] of the cascade of two LTI discrete-time systemswith impulse responses and is given by
][nh1][nh 2][nh1 ][nh 2
][][ nhnh 1= ][nh 2][nh1 *
][nh1 ][nh 2
][nh2][][ nhnh 1= *
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Cascade Connection
The ordering of the systems in the cascade has no effect onthe overall impulse response because of the commutative
property of convolution A cascade connection of two stable systems is stable A cascade connection of two passive (lossless) systems is
passive (lossless)
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Cascade Connection
An application is in the development of an inversesystem
If the cascade connection satisfies the relation
then the LTI system is said to be the inverse of
and vice-versa
][nh1 ][nh 2
][ nh 2][1 nh ][ n=
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Simple Interconnection Schemes
Consider the discrete-time system where
][nh 2
][nh1 +
+
][nh 4
][nh3
],1[5.0][][1 += nnnh
],1[25.0][5.0][2 = nnnh],[2][3 nnh =
][)5.0(2][4 nnhn =
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Simple Interconnection Schemes
Simplifying the block-diagram we obtain
][nh 2
][nh1 +
][][ 43 nhnh +
][nh1 +])[][(][
432 nhnhnh +
*
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Simple Interconnection Schemes
Overall impulse response h[n] is given by
Now ,
Finally
][][][][][ nhnhnhnhnh 42321 ++=])[][(][][][ nhnhnhnhnh 4321 ++= *
* *
][2])1[][(][][41
21
32 nnnnhnh =]1[][
21 = nn
* *
][][]1[][]1[][][21
21 nnnnnnnh =++=
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ESD2521Cl ifi ti f LTI Di t Ti
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Classification of LTI Discrete-TimeSystems
Based on Impulse Response Length - If the impulse response h[n] is of finite length, i.e.,
then it is known as a finite impulse response (FIR) discrete-time system The convolution sum description here is
2121 ,and for 0][ N N N n N nnh
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Classification of LTI Discrete-TimeSystems
The output y[n] of an FIR LTI discrete-time system can becomputed directly from the convolution sum as it is a finitesum of products
Examples of FIR LTI discrete-time systems are the
moving-average system and the linear interpolators
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Classification of LTI Discrete-TimeSystems
If the impulse response is of infinite length, then itis known as an infinite impulse response (IIR)discrete-time system
The class of IIR systems we are concerned with in
this course are characterized by linear constantcoefficient difference equations
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Classification of LTI Discrete-Time Systems
Example - The discrete-time accumulatordefined by
is an IIR system
][]1[][ n xn yn y +=
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Classification of LTI Discrete-TimeSystems
Based on the Output Calculation Process Nonrecursive System - Here the output can be
calculated sequentially, knowing only the presentand past input samples
Recursive System - Here the output computationinvolves past output samples in addition to the
present and past input samples
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Correlation of Signals
Definitions A measure of similarity between a pair of energy signals, x[n]
and y[n], is given by the cross-correlation sequencedefined by
The parameter called lag, indicates the time-shift between the pair of signals
][ xyr
...,,,],[][][ 210 ==
=n xy n yn xr
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Correlation of Signals
There are applications where it is necessary to compare one reference
signal with one or more signals to determine the similarity between the pair and to determine additional information based on the similarity
For example, in digital communications, a set of data symbols arerepresented by a set of unique discrete-time sequences
If one of these sequences has been transmitted, the receiver has to
determine which particular sequence has been received by comparingthe received signal with every member of possible sequences from theset
Similarly, in radar and sonar applications, the received signal reflectedfrom the target is a delayed version of the transmitted signal and by
measuring the delay, one can determine the location of the target The detection problem gets more complicated in practice, as often the
received signal is corrupted by additive ransom noise
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Correlation of Signals
y[n] is said to be shifted by samples to the right with respectto the reference sequence x[n] for positive values of , andshifted by samples to the left for negative values of
The ordering of the subscripts xy in the definition of
specifies that x[n] is the reference sequence which remainsfixed in time while y[n] is being shifted with respect to x[n]
][ xyr
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Correlation of Signals
If y[n] is made the reference signal and shift x[n] withrespect to y[n], then the corresponding cross-correlationsequence is given by
Thus, is obtained by time-reversing
= =
n yx n xn yr ][][][
][][][ =+=
= xym r m xm y
][ yxr ][ xyr
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Modern Communication System
Channelencoder
Channeldecoder
Interelaver
Deinterleaver
Modulator Tx filter
Rx filter Equalizer
Synchronizer
Demodulator
Higher-layer Networkprotocols
Analogfrontend
Function View
RISCcore
DSPcore
Hardwiredsignal processing /
channel codingDigital Circuit
I-RAM
D-RAM0
D-cache
I-cache
D-RAM1
DMAcontroller
peripheralbus
Logicsfor other
peripheral
SDRAMcontroller
PCMCIAbus
PLL
Chip Archi tecture View
Equalizer
Viterbidecoder
Digitalfilters
...
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Filtering Example
Signals are usually a mix of usefulinformation and noise
How do we extract the useful information?
Filtering is one way
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Filtering Example
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Transform Example
ki
Can you say which is 1 / # by looking at them? If not, go to frequency domain Another way to look at signals Done using transforms
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Transform Example
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Transform Equations
Discrete Fourier Transform x Time domain signal X Frequency domain representation of x
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Correlation Equation
Correlation x Transmitted signal y Received signal r xy- Correlation coefficients
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Amplification and Conditioning
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Amplification and Conditioning The voltage from a signal sensor is very small in magnitude. A microphone may
produce voltages of the order of 10 -6 volts. Similarly for ECG sensors,vibration sensors etc.
Prior to recording the signal or reproducing with an actuator an amplifier shouldsignal condition by linearly amplifying the signal by an appropriate factor.
The above amplifier adds 60dB of gain (20log101000 = 60)
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Amplifier Distortion
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Amplifier Distortion An amplifier which introduces unwanted artifacts, is said to be nonlinear and
is, of course, very undesirable as it may mask signal components of interest.
The above amplifier is non-linear and actually outputsthe input signal plus a 3rd order harmonic:
Unlike noise it is essentially impossible to remove theeffects of distortion. Therefore we try to minimize the
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Sig l d N i
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Signals and Noise
Most acquired signals are corrupted by some level of noise whichcan cause information to be lost.
Signal processing techniques are often used in an attempt toremove or attenuate noise.
Most noise can be considered as additive (linear superposition)which can be address by linear filtering techniques.
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The Noise/Distortion Chain
Consider the various levels of noise and distortion added in a
digital mobile communications link:
DSP must minimize the amount of noise/distortion inputto the chain, and where possible attenuate other sources.
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Sig l t N i R ti
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Signal to Noise Ratio
Taking the logarithm of the linear signal power to noise power
ratio (SNR) and multiplying by 10 gives the measure ofdecibels or dBs.
Recalling that Power is
Very low quality telephone line
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Generic Analogue Signal Processing
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Generic Analogue Signal Processing In general an analogue signal processing system can be defined as a system
that senses a signal to produce an analogue voltage, process this voltage,
and reproduce the signal to its original analogue form.
A public address system is an example of an analogue signal processingsystem:
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A Generic Input/Output DSP System
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A Generic Input/Output DSP System
A single input, single output DSP system has the followingcomponents:
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Generic Analogue Communications
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Generic Analogue Communications
For most base band telecommunications a voltage signal is
transmitted over a cable.
A simple example is a telephone . The acoustic signal isconverted to a voltage which is then directly transmittedover a twisted pair of wires to be received at a remotelocation.
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PEMPESD2521Analogue to Digital Converter (ADC)
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Analogue to Digital Converter (ADC)
An ADC is a device that can convert a voltage to a binary
number, according to its specific input-output characteristic.
The number of digital samples converted per second is defined by the sampling rate of the converter, f s Hz.
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g g ( )
A DAC is a device that can convert binary numbers to voltages,
according to its specific input-output characteristic.
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Signal Conditioning
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Signal Conditioning
Note that prior to a signal being input to an ADC, an amplifierwill be required to ensure that the full voltage range of theADC is used this is referred to as signal conditioning.
For the above ADC with a maximum input and output of 2volts we would require that the input signal to the ADC has asimilar range:
Depending on the output actuator, an amplifier, or at least a buffer amplifier will be required.
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Sampling
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Sampling
The speed at which an ADC generates binary numbers is calledthe sampling rate or sampling frequency f s
The time between samples is called the sampling period, t s:
Sampling frequency is quoted in samples per second, or simply as
Hertz (Hz). The actual sampling rate will depend on parameters of theapplication.
This may vary from:10s of Hz for control systems,
100s of Hz for biomedical,1000s of Hz for audio applications,1,000,000s of Hz for digital radio front ends .
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Sampling an Analogue Signal
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Sampling an Analogue Signal
After signal conditioning the ADC can produce binary number
equivalents of the input voltage. If the ADC has finite precision due to a limited no. of discretelevels then there may be a small error associated with eachsample.
The quantization step size is 0.0625 volts. If an 5 bit ADC is used,then the max/min voltage input is approx 0.0625 x 16 = 1 volt.
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Reproducing an Analogue Signal
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Reproducing an Analogue Signal
Using a DAC at an appropriate sampling rate, we can
reproduce an analogue signal:
Note that the output is a little steppy caused by the zeroorder hold (step reconstruction);....this artifact can however be removed with a reconstructionfilter.
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First Order Hold
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First Order Hold Alternatively a first order hold could be attempted in the
DAC. Here the voltage between two discrete samples isapproximated by a straight line.
A first order apparently produces a more accuratereproduction of the analogue signal. However implementationof a circuit to perform interpolation is not trivial and turns outnot to be necessary.
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Binary Data Word Lengths
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Binary Data Word Lengths
Data word lengths for DSP applications, typically:
Fixed Point Word lengths: Dynamic Range8 bits 128 to +127 20 log 2 8 48 dB16 bits 32768 to +32767 20 log 2 16 96 dB24 bits 8388608 to +8388607 20log2 24 154 dB
Floating Point Word lengths (for arithmetic only):32 bits (10 38 to +10 38)(24 bit mantissa, 8 bit exponent)
Note that data input from an ADC, or output to a DAC will always be fixed
point, although the internal DSP computation may be floating point.
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Sampling Too Fast ?
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Sampling Too Fast ? Sampling at f s = 800Hz, i.e. 8 samples per period:
Appears to be a reasonable sampling rate.
Sampling at f s = 3000Hz, i.e. 30 samples per period:
Perhaps higher than necessary sampling rate
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Sampling Too Slow
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Sampling Too Slow
Sampling at f s = 100Hz, i.e. 1 sample per period:
Signal interpreted as DC!
Sampling at f s = 100Hz, i.e. 1 sample per period:
Most of the signal features are missed
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Suitable Sampling Rate
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Suitable Sampling Rate
From inspection of the above 100Hz digital waveforms at the four
different sample rates:
f s = 800Hz seems a reasonable sampling rate; f s = 3000Hz is perhaps higher than necessary;
f s = 100Hz is too low and fails to correctly sample thewaveform, and loses the signal parameter information; f s = 80Hz is too low and fails completely
From mathematical theory the minimum sampling rate to retain allinformation is: greater than 2 x f max
where f max is the maximum frequency component of a baseband, bandlimited signal.
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Signal Frequency Range Terminology
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Signal Frequency Range Terminology
Nyquist frequency/rate: The Nyquist frequency, f n is identified astwice the maximum frequency component present in a signal.
Baseband: The lowest signal frequency present is around 0 Hz:
f l = lowest freq f h = highest freq f b = f h f l
Bandlimited: For all frequencies in the signal f h < f < f l
f b = Bandwidth
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Aliasing
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as g
When a (baseband) signal is sampled at a frequency below the Nyquist rate, then we lose the signal frequency informationand aliasing is said to have occured.
Aliasing can be illustrated by sampling a sine wave at below the Nyquist rate and then reconstructing. We note that it appears asa sine wave of a lower frequency (aliasing - cf. impersonating) .
Consider again sampling the 100Hz sine wave at 80Hz:
Reconstructed signal has a freq. of f s - f signal = 20Hz
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Aliasing Example
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Aliasing Example
Consider the output from the following three
systems:
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Aliased Spectra
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Aliased Spectra
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Anti Alias Filter
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Prior to the analogue to digital converter (ADC) all frequencies above f s/2must be blocked or they will be interpreted as lower frequencies, i.ealiasing.
The anti-alias filter is analogue (ideally a brick wall filter ),cutting off just before f s /2 Hz.
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Reconstruction Filter
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The analogue reconstruction filter at the output of a DAC removes the basebandimage high frequencies present in the signal (in the form of the steps between the
discrete levels).
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Zero Order Hold (ZOH)
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( ) Note that the operation of zero order hold can be interpreted as a
simple reconstructing frequency filtering operation:
The step reconstruction therefore causes a droop near f s/2.
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Anti-Alias and Reconstruction
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Anti-alias and reconstructions filters are analogue, i.e. made fromresistors, capacitors, amplifiers, even inductors.
Ideally they are both very sharp cut off filters at frequency f s/2. In practice the roll off will be between 6dB/octave (a simple resistorand capacitor) to 96dB/octave (a 16th order filter).
Steeper roll-off is more expensive, but clearly for many applications,
good analogue filters are essential. In a DSP system the accurately trimmed analogue filters could
actually be more costly than the other DSP components: i.e. DSP processor, ADC, DAC, memory etc.
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Perfect Nyquist Sampling
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yq p g The Nyquist sampling theorem states that a (baseband) signal should sampled at
greater than twice the maximum frequency component present in the signal:
f s > 2 * f max The sampled signal can then be perfectly reconstructed to the original analogue
signal with no added noise or distortion.
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ADC Sampling Error
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Perfect signal reconstruction assumes that sampled data values areexact (i.e. infinite precision real numbers). In practice they are not,as an ADC will have a number of discrete levels.
The ADC samples at the Nyquist rate, and the sampled data valueis the closest (discrete) ADC level to the actual value:
p g
Hence every sample has a small quantization error.
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ADC Sampling Error
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Assume an ADC or quantizer has 5 bits of resolution andmaximum/minimum voltage swing of +1 and -1 volts. The input/output
characteristic is shown below:
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If the smallest step size of a linear ADC is q volts , then the error of any onesample is at worst q/2 volts.
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The quantization error is straightforward to calculate from:
q = V max / 2 N 1
where N is the number of bits in the converter.
The dynamic range of an bit converter is often quoted in dBs:
Dynamic Range=20log 102 N =20 N log 102=6.02 N
Therefore an 8 bit converter has a range of
Binary 10000000 to 01111111, or in decimal -128 to 127 has a dynamic rangeof approximately 48 dB.
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The actual ADC can be represented by a sampler and a quantizer:
The quantization error of each sample is in the range and we can model thequantizer as a linear additive noise source.
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PEMPESD2521Timing Jitter Error
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Note that when a signal is sampled there may be some jitter on thesampling clock which will cause additional sample error.
With jitter each sampling instant may be slightly offset, andtherefore the sample value obtained and sent to the DSP will bein error.
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Computation Algorithm Examples
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Computation Algorithm Examples
LMS Algorithms
Some simplified algorithms
)()()(
)()()( 1
0
n ynd ne
nwk n xn y M
k k
== =
)()()()1( k n xnenwnw k k +=+
)())((sign)()1( k n xnenwnw k k +=+
))((sign))((sign)()1( k n xnenwnw k k +=+ ( ) )())((sign2)()1( )(log 2 k n xnenwnw nek k +=+ +
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How is Signal Processed
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How is Signal Processed Analog-Digital Conversion
Digital representation Digital signal processing
anti-aliasLPF
sample&
hold
quantizer +
coder latch
Ts : sampling period
pulse stream
enableBw = 1/(2Ts)
waveform
analogLPF
reg Amplitudemapper
waveform digitaldata
digitaldata
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ESD2521
Sampling and Quantization
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Sampling and Quantization
Sample and hold
Quantization
0-Ts
-2Ts
-3Ts
-4Ts T s
3Ts
4Ts
5Ts
6Ts
7Ts
8Ts
0-Ts
-2Ts
-3Ts
-4Ts T s
3Ts
4Ts
5Ts
6Ts
7Ts
8Ts1
23
4
56
78
910
-1-2-3
-4
-5
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ESD2521
Session Summary
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Sess o Su a y Functional Architecture Hardware Architecture
System-level algorithms vs Hardware computation-efficient algorithms
Trade-off and System performance and Hardwarecomplexity (cost)
Signal processing is focused on efficient implementationof integrated circuit.
Signals can be classified as continuous-time and discretetime signals.
A system is BIBO stable if its impulse response isabsolutely summable.
A response of an LTI system is the convolution of its