Session 2 TD Filt

8/13/2019 Session 2 TD Filt

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M.S. Ramaiah School of Advanced Studies, Bengaluru 1

Session 2 : Review On Signals

Session delivered by:

Chandan N.


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M.S. Ramaiah School of Advanced Studies, Bengaluru 2

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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M.S. Ramaiah School of Advanced Studies, Bengaluru

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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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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S 2 21


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Simplifying the block-diagram we obtain

][nh 2

][nh1 +

][][ 43 nhnh +

][nh1 +])[][(][

432 nhnhnh +

*

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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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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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ESD2521l f f


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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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ESD2521Cl ifi i f LTI Di Ti


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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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S 5



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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ESD2521C l ti f Si l


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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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C l i f Si l


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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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Th N i /Di t ti Ch i


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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

Audio Cassette Deck65

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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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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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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

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-1-2-3

-4

-5

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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

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Session 2 TD Filt