Properties Convolution linear system's characteristics completely

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Properties Convolution linear system's characteristics completely

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Properties Convolution
linear system's characteristics completely specified system's impulse response, governed mathematics convolution. This basis many signal processing techniques. example: Digital filters created designing appropriate impulse response. Enemy aircraft detected with radar analyzing measured impulse response. Echo suppression long distance telephone calls accomplished creating impulse response that counteracts impulse response reverberation. list goes This chapter expands properties usage convolution several areas. First, several common impulse responses discussed. Second, methods presented dealing with cascade parallel combinations linear systems. Third, technique correlation introduced. Fourth, nasty problem with convolution examined, computation time unacceptably long using conventional algorithms computers.
Common Impulse Responses
Delta Function simplest impulse response nothing more that delta function, shown Fig. 7-1a. That impulse input produces identical impulse output. This means that signals passed through system without change. Convolving signal with delta function results exactly same signal. Mathematically, this written:
EQUATION delta function identity convolution. signal convolved with delta function left unchanged.
This property makes delta function identity convolution. This analogous zero being identity addition being identity multiplication first glance, this type system
Scientist Engineer's Guide Digital Signal Processing
seem trivial uninteresting. Such systems ideal data storage, communication measurement. Much concerned with passing information through systems without change degradation. Figure 7-1b shows slight modification delta function impulse response. delta function made larger smaller amplitude, resulting system amplifier attenuator, respectively. equation form, amplification results greater than one, attenuation results less than one:
EQUATION system that amplifies attenuates scaled delta function impulse response. this equation, determines amplification attenuation.
k*[n]
impulse response Fig. 7-1c delta function with shift. This results system that introduces identical shift between input output signals. This could described signal delay, signal advance, depending direction shift. Letting shift represented parameter, this written equation:
EQUATION relative shift between input output signals corresponds impulse response that shifted delta function. variable, determines amount shift this equation.
*[n%
Science engineering filled with cases where signal shifted version another. example, consider radio signal transmitted from remote space probe, corresponding signal received earth. time takes radio wave propagate over distance causes delay between transmitted received signals. biology, electrical signals adjacent nerve cells shifted versions each other, determined time takes action potential cross synaptic junction that connects two. Figure 7-1d shows impulse response composed delta function plus shifted scaled delta function. superposition, output this system input signal plus delayed version input signal, i.e., echo. Echoes important many applications. addition echoes part making audio recordings sound natural pleasant. Radar sonar analyze echoes detect aircraft submarines. Geophysicists echoes find oil. Echoes also very important telephone networks, because want avoid them.
Chapter Properties Convolution
Identity delta function identity convolution. Convolving signal with delta function leaves signal unchanged. This goal systems that transmit store signals.
Amplitude
Sample number
Amplification Attenuation Increasing decreasing amplitude delta function forms impulse response that amplifies attenuates, respectively. This impulse response will amplify signal 1.6.
Amplitude
Sample number
Shift Shifting delta function produces corresponding shift between input output signals. Depending direction, this called delay advance. This impulse response delays signal four samples.
Amplitude
Sample number
Echo delta function plus shifted scaled delta function results echo being added original signal. this example, echo delayed four samples amplitude original signal.
Amplitude
Sample number
FIGURE Simple impulse responses using shifted scaled delta functions.
Calculus-like Operations Convolution change discrete signals ways that resemble integration differentiation. Since terms "derivative" "integral" specifically refer operations continuous signals, other names given their discrete counterparts. discrete operation that mimics first derivative called first difference. Likewise, discrete form integral called
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running sum. also common hear these operations called discrete derivative discrete integral, although mathematicians frown when they hear these informal terms used. Figure shows impulse responses that implement first difference running sum. Figure shows example using these operations. 73a, original signal composed several sections with varying slopes. Convolving this signal with first difference impulse response produces signal Fig. 7-3b. Just with first derivative, amplitude each point first difference signal equal slope corresponding location original signal. running inverse operation first difference. That convolving signal (b), with running sum's impulse response, produces signal (a). These impulse responses simple enough that full convolution program usually needed implement them. Rather, think them alternative mode: each sample output signal weighted samples from input. instance, first difference calculated:
EQUATION Calculation first difference. this relation, original signal, first difference.
That each sample output signal equal difference between adjacent samples input signal. instance, y[40] x[40] x[39] should mentioned that this only define discrete derivative. Another common method define slope symmetrically around point being examined, such y[n] x[n% x[n&
First Difference This discrete version first derivative. Each sample output signal equal difference between adjacent samples input signal. other words, output signal slope input signal.
Amplitude
Sample number
Running running discrete version integral. Each sample output signal equal samples input signal left. Note that impulse response extends infinity, rather nasty feature.
Amplitude
Sample number
FIGURE Impulse responses that mimic calculus operations.
Chapter Properties Convolution
Original signal
Amplitude
-1.0
FIGURE Example calculus-like operations. signal first difference signal (a). Correspondingly, signal running signal (b). These processing methods used with discrete signals same differentiation integration used with continuous signals.
-2.0
Sample number
First Difference
Running
First difference
Amplitude
-0.1 -0.2
Sample number
Using this same approach, each sample running calculated summing points original signal left sample's location. instance, y[n] running x[n] then sample y[40] found adding samples x[0] through x[40] Likewise, sample y[41] found adding samples x[0] through x[41] course, would very inefficient calculate running this manner. example, y[40] already been y[41] y[41] x[41] y[40] equation form:
EQUATION Calculation running sum. this relation, original signal, running sum.
Relations this type called recursion equations difference equations. will revisit them Chapter now, important idea understand that these relations identical convolution using impulse responses Fig. 7-2. Table provides computer programs that implement these calculus-like operations.
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'Calculation First Difference Y[0] N%-1 Y[I%] X[I%] Y[I%-1] NEXT 'Calculation running Y[0] X[0] N%-1 Y[I%] Y[I%-1] X[I%] NEXT
Table Programs calculating first difference running sum. original signal held processed signal (the first difference running sum) held Both arrays from N%-1.
Low-pass High-pass Filters design digital filters covered detail later chapters. now, satisfied understand general shape low-pass high-pass filter kernels (another name filter's impulse response). Figure shows several common low-pass filter kernels. general, low-pass filter kernels composed group adjacent positive points. This results each sample output signal being weighted average many adjacent points from input signal. This averaging smoothes signal, thereby removing highfrequency components. shown sinc function (c), some low-pass filter kernels include negative valued samples tails. Just analog electronics, digital low-pass filters used noise reduction, signal separation, wave shaping, etc.
Exponential
Square pulse
Amplitude
-0.1
Amplitude
Sample number
-0.1
Sample number
FIGURE Typical low-pass filter kernels. Low-pass filter kernels formed from group adjacent positive points that provide averaging (smoothing) signal. discussed later chapters, each these filter kernels best particular purpose. exponential, (a), simplest recursive filter. rectangular pulse, (b), best reducing noise while maintaining edge sharpness. sinc function (c), curve form: sin(x)/(x) used separate band frequencies from another.
Sinc
Amplitude
-0.1
Sample number
Chapter Properties Convolution
1.50
Exponential
1.00
Square pulse
Amplitude
Amplitude
0.50
0.00
-0.5
Sample number
-0.50
Sample number
1.50
Sinc
FIGURE Typical high-pass filter kernels. These formed subtracting corresponding lowpass filter kernels Fig. from delta function. distinguishing characteristic high-pass filter kernels spike surrounded many adjacent negative samples.
1.00
Amplitude
0.50
0.00
-0.50
Sample number
cutoff frequency filter changed making filter kernel wider narrower. low-pass filter gain (zero frequency), then points impulse response must equal one. illustrated (c), some filter kernels theoretically extend infinity without dropping value zero. actual practice, tails truncated after certain number samples, allowing represented finite number points. else could stored computer? Figure shows three common high-pass filter kernels, derived from corresponding low-pass filter kernels Fig. 7-4. This common strategy filter design: first devise low-pass filter then transform what need, high-pass, band-pass, band-reject, etc. understand low-pass high-pass transform, remember that delta function impulse response passes entire signal, while low-pass impulse response passes only lowfrequency components. superposition, filter kernel consisting delta function minus low-pass filter kernel will pass entire signal minus low-frequency components. high-pass filter born! shown Fig. 7-5, delta function usually added center symmetry, sample zero filter kernel symmetrical. High-pass filters have zero gain (zero frequency), achieved making points filter kernel equal zero.
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Causal
Causal
Amplitude
-0.1
Amplitude
Sample number
-0.1
Sample number
Noncausal
FIGURE Examples causal signals. impulse response, signal, said causal negative numbered samples have value zero. Three examples shown here. noncausal signal with finite number points turned into causal signal simply shifting.
Amplitude
-0.1
Sample number
Causal Noncausal Signals Imagine simple analog electronic circuit. apply short pulse input, will response output. This kind cause effect that universe based thing definitely know: effect must happen after cause. This basic characteristic what call time. compare this system that changes input signal into output signal, both stored arrays computer. this mimics real world system, must follow same principle causality real world does. example, value sample number eight input signal only affect sample number eight greater output signal. Systems that operate this manner said causal. course, digital processing doesn't necessarily have function this way. Since both input output signals arrays numbers stored computer, input signal values affect output signal values. shown examples Fig. 7-6, impulse response causal system must have value zero negative numbered samples. Think this from input side view convolution. causal, impulse input signal sample number must only affect those points output signal with sample number greater. common usage, term causal applied signal where negative numbered samples have value zero, whether impulse response not.
Chapter Properties Convolution
Zero phase
Linear phase
Amplitude
-0.1
Amplitude
Sample number
-0.1
Sample number
FIGURE Examples phase linearity. Signals that have left-right symmetry said linear phase. axis symmetry occurs sample number zero, they additionally said zero phase. linear phase signal transformed into zero phase signal simply shifting. Signals that have leftright symmetry said nonlinear phase. confuse these terms with linear linear systems. They completely different concepts.
Nonlinear phase
Amplitude
-0.1
Sample number
Zero Phase, Linear Phase, Nonlinear Phase shown Fig. 7-7, signal said zero phase left-right symmetry around sample number zero. signal said linear phase left-right symmetry, around some point other than zero. This means that linear phase signal changed into zero phase signal simply shifting left right. Lastly, signal said nonlinear phase does have left-right symmetry. probably thinking that these names don't seem follow from their definitions. What does phase have with symmetry? answer lies frequency spectrum, will discussed more detail later chapters. Briefly, frequency spectrum signal composed parts, magnitude phase. frequency spectrum signal that symmetrical around zero phase that zero. Likewise, frequency spectrum signal that symmetrical around some nonzero point phase that straight line, i.e., linear phase. Lastly, frequency spectrum signal that symmetrical phase that straight line, i.e., nonlinear phase. special note about potentially confusing terms: linear nonlinear phase. What does this have concept system linearity discussed previous chapters? Absolutely nothing! System linearity broad concept
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that nearly based (superposition, homogeneity, additivity, etc). Linear nonlinear phase mean that phase not, straight line. fact, system must linear even that phase zero, linear, nonlinear.
Mathematical Properties
Commutative Property commutative property convolution expressed mathematical form:
EQUATION commutative property convolution. This states that order which signals convolved exchanged.
words, order which signals convolved makes difference; results identical. shown Fig. 7-8, this strange meaning system theory. linear system, input signal system's impulse response exchanged without changing output signal. This interesting, usually doesn't have physical meaning. input signal impulse response very different things. Just because mathematics allows something, doesn't mean that makes sense example, suppose make: $10/hour 2,000 hours/year $20,000/year. commutative property multiplication provides that make same annual salary only working hours/year $2000/hour. Let's convince your boss that this meaningful! spite this, commutative property sees great manipulating equations, just ordinary algebra.
a[n]
b[n]
y[n]
THEN
b[n]
a[n]
y[n]
FIGURE commutative property system theory. commutative property convolution allows input signal impulse response system exchanged without changing output. While interesting, this usually physical significance. signal appearing inside box, such b[n] a[n] this figure, represent impulse response system).
Chapter Properties Convolution
Associative Property possible convolve three more signals? answer yes, associative property describes how: convolve signals produce intermediate signal, then convolve intermediate signal with third signal. associative property provides that order convolutions doesn't matter. equation:
EQUATION associative property convolution describes three more signals convolved.
associative property used system theory describe cascaded systems behave. shown Fig. 7-9, more systems said cascade output system used input next system. From associative property, order systems rearranged without changing overall response cascade. Further, number cascaded systems replaced with single system. impulse response replacement system found convolving impulse responses original systems.
x[n]
h1[n] h2[n]
y[n]
THEN
x[n]
h2[n] h1[n]
y[n]
ALSO
x[n]
h1[n] h2[n]
y[n]
FIGURE associative property system theory. associative property provides important characteristics cascaded linear systems. First, order systems rearranged without changing overall operation cascade. Second, more systems cascade replaced single system. impulse response replacement system found convolving impulse responses stages being replaced.
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Distributive Property equation form, distributive property written:
EQUATION distributive property convolution describes parallel systems analyzed.
distributive property describes operation parallel systems with added outputs. shown Fig. 7-10, more systems share same input, x[n] have their outputs added produce y[n] distributive property allows this combination systems replaced with single system, having impulse response equal impulse responses original systems.
h1[n]
x[n]
h2[n]
y[n]
THEN
x[n]
h1[n] h2[n]
y[n]
FIGURE 7-10 distributive property system theory. distributive property shows that parallel systems with added outputs replaced with single system. impulse response replacement system equal impulse responses original systems.
Transference between Input Output Rather than being formal mathematical property, this thinking about common situation signal processing. illustrated Fig. 7-11, imagine linear system receiving input signal, x[n] generating output signal, y[n] suppose that input signal changed some linear way, resulting input signal, which will call 3[n] This results output signal, 3[n] question does change
Chapter Properties Convolution
input signal relate change output signal? answer output signal changed exactly same linear that input signal changed. example, input signal amplified factor two, output signal will also amplified factor two. derivative taken input signal, derivative will also taken output signal. input filtered some way, output will filtered identical manner. This easily proven using associative property.
x[n]
h[n]
y[n]
Linear Change
Same Linear Change
THEN
h[n]
FIGURE 7-11 Tranference between input output. This thinking about common situation signal processing. linear change made input signal results same linear change being made output signal.
Central Limit Theorem Central Limit Theorem important tool probability theory because mathematically explains Gaussian probability distribution observed commonly nature. example: amplitude thermal noise electronic circuits follows Gaussian distribution; cross-sectional intensity laser beam Gaussian; even pattern holes around dart board bull's Gaussian. simplest form, Central Limit Theorem states that Gaussian distribution results when observed variable many random processes. Even component processes have Gaussian distribution, them will. Central Limit Theorem interesting implication convolution. pulse-like signal convolved with itself many times, Gaussian produced. Figure 7-12 shows example this. signal
3.00
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18.0
x[n]
2.00 12.0
x[n] x[n]
Amplitude
1.00
Amplitude
0.00
-1.00
-6.0
Sample number
Sample number
1500
x[n] x[n]
x[n] x[n]
FIGURE 7-12 Example convolving pulse waveform with itself. Central Limit Theorem shows that Gaussian waveform produced when arbitrary shaped pulse convolved with itself many times. Figure example pulse. (b), pulse convolved with itself once, begins appear smooth regular. (c), pulse convolved with itself three times, closely approximates Gaussian.
1000
Amplitude
-500
Sample number
irregular pulse, purposely chosen very unlike Gaussian. Figure shows result convolving this signal with itself time. Figure shows result convolving this signal with itself three times. Even with only three convolutions, waveform looks very much like Gaussian. mathematics jargon, procedure converges Gaussian very quickly. width resulting Gaussian (i.e., 2-8) equal width original pulse (expressed 2-7) multiplied square root number convolutions.
Correlation
concept correlation best presented with example. Figure 7-13 shows elements radar system. specially designed antenna transmits short burst radio wave energy selected direction. propagating wave strikes object, such helicopter this illustration, small fraction energy reflected back toward radio receiver located near transmitter. transmitted pulse specific shape that have selected, such triangle shown this example. received signal will consist parts: shifted scaled version transmitted pulse, random noise, resulting from interfering radio waves, thermal noise electronics, etc. Since radio signals travel known rate, speed
Chapter Properties Convolution
light, shift between transmitted received pulse direct measure distance object being detected. This problem: given signal some known shape, what best determine where signal occurs another signal. Correlation answer. Correlation mathematical operation that very similar convolution. Just with convolution, correlation uses signals produce third signal. This third signal called cross-correlation input signals. signal correlated with itself, resulting signal instead called autocorrelation. convolution machine presented last chapter show convolution performed. Figure 7-14 similar
FIGURE 7-13 elements radar system. Like other echo location systems, radar transmits short pulse energy that reflected objects being examined. This makes received waveform shifted version transmitted waveform, plus random noise. Detection known waveform noisy signal fundamental problem echo location. answer this problem correlation.
TRANSMIT
RECEIVE
Transmitted amplitude
-100
Sample number time)
Received amplitude
-0.1
Sample number time)
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illustration correlation machine. received signal, x[n] cross-correlation signal, y[n] fixed page. waveform looking for, t[n] commonly called target signal, contained within correlation machine. Each sample y[n] calculated moving correlation machine left right until points sample being worked Next, indicated samples from received signal fall into correlation machine, multiplied corresponding points target signal. these products then moves into proper sample crosscorrelation signal. amplitude each sample cross-correlation signal measure much received signal resembles target signal, that location. This means that peak will occur cross-correlation signal every target signal that present received signal. other words, value cross-correlation maximized when target signal aligned with same features received signal. What target signal contains samples with negative value? Nothing changes. Imagine that correlation machine positioned such that target signal perfectly aligned with matching waveform received signal. samples from received signal fall into correlation machine, they multiplied their matching samples target signal. Neglecting noise, positive sample will multiplied itself, resulting positive number. Likewise, negative sample will multiplied itself, also resulting positive number. Even target signal completely negative, peak cross-correlation will still positive. there noise received signal, there will also noise crosscorrelation signal. unavoidable fact that random noise looks certain amount like target signal choose. noise cross-correlation signal simply measuring this similarity. Except this noise, peak generated cross-correlation signal symmetrical between left right. This true even target signal isn't symmetrical. addition, width peak twice width target signal. Remember, cross-correlation trying detect target signal, recreate There reason expect that peak will even look like target signal. Correlation optimal technique detecting known waveform random noise. That peak higher above noise using correlation than produced other linear system. perfectly correct, only optimal random white noise). Using correlation detect known waveform frequently called matched filtering. More this Chapter correlation machine convolution machine identical, except small difference. discussed last chapter, signal inside convolution machine flipped left-for-right. This means that samples numbers: from right left. correlation machine this flip doesn't take place, samples normal direction.
Chapter Properties Convolution
x[n]
Sample number time)
t[n]
y[n]
Sample number time)
FIGURE 7-14 correlation machine. This flowchart showing cross-correlation signals calculated. this example, cross-correlation dashed moved left right that output points sample being calculated indicated samples from multiplied corresponding samples products added. correlation machine identical convolution machine (Figs. 6-9), except that signal inside dashed reversed. this illustration, only samples calculated where fully immersed
Since this signal reversal only difference between operations, possible represent correlation using same mathematics convolution. This requires preflipping signals being correlated, that left-for-right flip inherent convolution canceled. instance, when a[n] b[n] convolved produce c[n] equation written: comparison, cross-correlation a[n] b[n]
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written: That flipping left-for-right accomplished reversing sign index, i.e., Don't mathematical similarity between convolution correlation fool you; they represent very different procedures. Convolution relationship between system's input signal, output signal, impulse response. Correlation detect known waveform noisy background. similar mathematics only convenient coincidence.
Speed
Writing program convolve signal another simple task, only requiring lines code. Executing program more painful. problem large number additions multiplications required algorithm, resulting long execution times. shown programs last chapter, time-consuming operation composed multiplying numbers adding result accumulator. Other parts algorithm, such indexing arrays, very quick. multiply-accumulate basic building block DSP, will repeated several other important algorithms. fact, speed computers often specified long takes preform multiply-accumulate operation. signal composed samples convolved with signal composed samples, multiply-accumulations must preformed. This seen from programs last chapter. Personal computers 1990's requires about microsecond multiply-accumulation (100 Pentium using single precision floating point, Table 4-6). Therefore, convolving 10,000 sample signal with sample signal requires about second. process million point signal with 3000 point impulse response requires nearly hour. decade earlier (80286 MHz), this calculation would have required three days! problem excessive execution time commonly handled three ways. First, simply keep signals short possible integers instead floating point. only need convolution times, this will probably best trade-off between execution time programming effort. Second, computer designed DSP. microprocessors available with multiply-accumulate times only tens nanoseconds. This route plan perform convolution many times, such design commercial products. third solution better algorithm implementing convolution. Chapter describes very sophisticated algorithm called convolution. convolution produces exactly same result convolution algorithms presented last chapter; however, execution time dramatically reduced. signals with thousands samples, convolution hundreds times faster. disadvantage program complexity. Even familiar with technique, expect spend several hours getting program run.

Properties Convolution linear system's characteristics completely

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