Advanced Signals and Systems Discrete Systems Gerhard Schmidt - - PowerPoint PPT Presentation
Advanced Signals and Systems Discrete Systems Gerhard Schmidt Christian-Albrechts-Universitt zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory Digital Signal
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Gerhard Schmidt
Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-2
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Introduction Discrete signals and random processes Spectra Discrete systems Idealized linear, shift-invariant systems Hilbert transform State-space description and system realizations Generalizations for signals, systems, and spectra
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-3
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-4
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
The most general description of systems defines a system as on operator Typically we will use the following graphical description for systems Beyond the input and the output signal vector also internal signals can contribute to the system
with state variables that contain the complete information on the system state in a non-redundant form.
System
Basics:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-5
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
It is also possible to describe a system by means of its state-space description. Usually two equations are used for that purpose:
The output equation, that describes how the output signal vector is generated using
the current input vector and the current state vector
and the state equation, that describes how the new state vector is generated using the
current state vector and the current input vector
Basics (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-6
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Continuous, discrete, and digital systems One- and multi-dimensional systems Deterministic and stochastic systems Passivity Dynamic and memoryless systems Real and complex systems Causality Linearity Shift or time invariance Stability Discrete linear systems and their response to deterministic signals Stability of linear systems ...
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-7
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
In our notation of the last slides, the description of discrete systems respectively produce, upon a discrete input signal, only a discrete output signal. In contrast to that also continuous state-space descriptions exist. Since such descriptions are based on differential equations (instead of difference equations) the derivative of the state vector is utilized in the state equation:
Continuous, discrete, and digital systems:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-8
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Continuous, discrete, and digital systems (continued):
In discrete systems the state variables usually describe the contents of memory elements. In continuous systems usually energy-storing elements are described by the state variable. Some general comments:
A system taking a number of input sequences and producing a number of output
sequences (and ) may, of course, be generally seen as a computing system.
However, digital computers necessarily work with quantized (limited word length)
signals and variables. These digital systems are a special type of discrete systems.
Here (in this lecture) we will deal only with discrete systems – digital systems are
treated in the lecture “Advanced digital signal processing”.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-9
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
One-dimensional and multi-dimensional systems:
Within the introduction slides of this lecture we had already explained the difference between scalar and vector signals on the one hand and one- and multi-dimensional signals on the
For systems we can apply the same definition. However, as mentioned before, in this section we will focus first on one-dimensional systems – multi-dimensional extensions will be covered briefly at the end of this part.
E.g. brightness of a picture ( ) E.g. speed of an object (x, y, and z-direction)
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-10
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Deterministic and stochastic systems:
If is fixed (not necessarily constant) in all aspects (structure, parameters), then is
stochastic. Consequence: For a stochastic system, is stochastic even for a deterministic input signal (while for a deterministic system, is deterministic for a deterministic input signal)! We will focus in the remaining slides on deterministic systems.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-11
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Passivity:
In the second part of this lecture (“Signals and Stochastic Processes”) we have introduced the instantaneous or local energy: This quantity is used to determine passive systems. For a system with one input and one output the system is passive if the instantaneous power of the output signal is always smaller or equal than the instantaneous input power: For passive systems with more than one input or output we define passivity as
Please note that both definitions must be fulfilled for all !
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-12
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Passivity (continued):
As consequences we can conclude that …
… no signal sources are allowed inside a passive system and … if all input signals are zero up to a certain index
than also all output signals of a passive system must be zero
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-13
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Dynamic and memoryless systems:
Definition of a dynamic system: A system is called a dynamic system if does not only depend on , but also on . Consequences:
Dynamic systems must contain some “storage” of values beyond the index . The
storage may concern
values for (“left” of , for representing a time index this means
“before “, meaning storage of the past, i.e. memory in the usual sense).
values for (“right” of , for representing a time index this means
“after “, meaning storage of the future, which is possible if is not related to real-time). In contrast to that: Systems without storage or memory (“non-dynamic" systems) are called memoryless.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-14
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Real-valued and complex-valued systems:
If all elements of the input signal vector of a system are real and also all elements of the output signal vector are real then the system is called a real-valued (or real) system. Otherwise we call the system to be complex-valued (or simply complex).
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-15
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Causality:
Definition of a causal system: If depends only on , is called causal. If has the meaning of a time index, this means that the system output depends only on current and past inputs. In addition, we define: If depends only on , is called anti-causal. Finally, if both conditions are fulfilled at the same time, we define a system as non-causal: If depends on both and , is called non-causal. Consequences:
Causality commonly means that a reason or “cause” precedes naturally any
“reaction” – so, the name actually implies as a time index. In this case, “real systems” are usually causal – not necessarily in other cases!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-16
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Causality (continued):
Consequences (continued):
Memoryless systems are causal. This is a consequence of the definition of memoryless
systems.
Passive systems are causal. Again this follows from the definition of this system property. In literature, sometimes right-hand sided signals, with
are termed “causal signals”. Correspondingly, if we have a so-called left-hand sided signal is sometimes called an “anti-causal signal”, and a general two-sided signal is called a “non-causal signal”. These expressions are sometimes misleading and, thus, avoided here.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-17
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Linearity:
Linear systems are defined by the fact that the superposition theorem holds: This can be visualized by the following diagram:
… only in case of linearity!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-18
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Linearity (continued):
Remarks on the superposition theorem:
The superposition theorem includes, of course, any finite superposition. The theorem may even be extended to an infinite summation of “infinitely-dense“
contributions, i.e. to integrate on both sides (will be treated later on). General remarks:
Linear systems will be in the focus of this course. This does not mean that most “real-world systems” are linear (on the contrary,
most systems are non-linear).
For linear systems, however, a widely general theory is applicable. Therefore, it is often
tried to deal with actually non-linear systems after “linearization” (valid near some “point
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-19
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Shift or time invariance:
A system is called shift- or time-invariant if the order of a delay operator and the system operator can be exchanged: with Graphical interpretation:
Delay Delay … only if system is shift-invariant!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-20
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Shift or time invariance (continued):
Remarks:
Linearity and shift invariance are independent system characteristics. There are,
They are called LTI systems (LTI stands for linear and time invariant). For these LTI system a very powerful theory can be applied. Like linearity, shift or time invariance is important, but not given in general: many
systems are at least “slowly " time-variant (e.g., due to aging) – these systems are denoted as short-term or pseudo time invariant.
In the diagram on the last slide a basic delay operator appears. This operator can
be realized on a computer as a shift register.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-21
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Stability:
A system that reacts on a bounded input signal also with a bounded output signal is called a BIBO-stable system. For a BIBO-stable system with a single input and a single output this means, that for the system output will be also bounded For BIBO-stable systems with more than one input and/or output we get in the same manner: if all inputs are limited according to than also all output will be bounded
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-22
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Partner work:
Partner work – Please think about the following questions and try to find answers (first group discussions, afterwards broad discussion in the whole group).
How do you check if a system is linear? What can you say about the linearity of the
following systems: and ? …………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………….
What can you say about the time-invariance of the following systems:
and ? …………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………….
Please name a few (different) meanings of the term „system dynamic“. Which meaning
will we use in this lecture? …………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………….
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-23
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Continuous, discrete, and digital systems One- and multi-dimensional systems Deterministic and stochastic systems Passivity Dynamic and memoryless systems Real and complex systems Causality Linearity Shift or time invariance Stability Discrete linear systems and their response to deterministic signals Stability of linear systems …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-24
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Reactions on elementary signals Reaction of LTI systems on exponential signals Reaction on general deterministic signals Relations between the system descriptions Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-25
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Introduction:
We will discuss now a “more detailed” system description in terms of the system reaction (the output signal) on a specific input signal . The question will be: What does the operator do with ? In the following we will assume only one input and one output signal – the general case of more than one input and output will be treated later on. Since every different produces a specific , it is necessary to reduce our considerations to very few input signals. The aims will be now
Find few standard-signal responses and apply these to also describe responses to general signals!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-26
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reactions on elementary signals:
We will assume now a linear system according to the following picture: Next we define the system responses to the following four input sequences:
an impulse sequence a unit step sequence
Linear system System response to an impulse Only in case of shift invariance System response to a step sequence
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Slide IV-27
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reactions on elementary signals (continued):
System responses to four input sequences (continued):
an harmonic exponential sequence a general exponential sequence
The following names have been established for the individual system descriptions:
Impulse response: or shortly Step response: Frequency response: Transfer function:
Only in case of shift invariance, proof for the “behavior” will come later.
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Slide IV-28
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction of LTI systems on exponential signals:
If we assume that a system is linear and time invariant (LTI) than the reaction of the system on an exponential (either harmonic or general) sequence is also an exponential sequence with the same frequency and the same decaying/increasing time constants. This can be proven as follows: Given is a general exponential input sequence (this includes a harmonic exponential sequence): The reaction of an LTI system is Next, we have a look on the system output to shifted input:
… due to linearity this can be changed to ... … inserting and simplifying ...
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Slide IV-29
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction of LTI systems on exponential signals (continued):
The system output to this shifted sequence is Due to the shift-invariance of the system we obtain also Thus, we obtain for the output signal of the LTI system
… inserting the definition of the input sequence ... … assuming linearity of the system ... … inserting the definition of the non-shifted output signal ...
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-30
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction of LTI systems on exponential signals (continued):
The only signal sequence that fulfills the condition is again a general exponential sequence Proof: As a result we get the LTI system response to a general exponential sequence
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-31
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction of LTI systems on exponential signals (continued):
The complex amplitude at the output is, in general, different from that at the input , and its change by the system depends on the choice of . The ratio of the two complex amplitudes is called transfer function and is defined as The derivation of the last few slides includes also the case of harmonic exponential input sequences Here the amplitude ratio is called frequency response and we get
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-32
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction of LTI systems on exponential signals (continued):
The derivation of the last slides includes also sinusoidal excitation sequences. They can be generated by taking the real part of a complex harmonic exponential sequence: If we compute also the real part of the output sequence, we obtain Again, this is a sinusoidal sequence with the same frequency. Only the phase and the (real) amplitude are changed.
Exponentials (including sinusoids) are reproduced by LTI systems. Such sequences are called “eigenfunction” of LTI systems. The reproduction property is the reason why spectral decomposition into such components are applied so widely!
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Slide IV-33
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction on general deterministic signals:
In the previous sections we analyzed the response of an LTI system to the four basic input sequences A general input sequence can be decomposed as a weighted sum of these basic sequences (see last part of this lecture about spectral representations). Due to linearity we can now describe the LTI system output as a weighted sum of system reactions to the basis sequences: Descriptions by means of the impulse response:
Input sequence: Output sequence:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-34
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction on general deterministic signals (continued):
Descriptions by means of the step response:
Input sequence:
with
Output sequence:
Both, for the impulse response and for the step response, the convolution is a basic operation – a so-called “kernel” operation!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-35
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Reaction on general deterministic signals (continued):
Descriptions by means of the frequency response:
Input sequence: Output sequence:
Descriptions by means of the transfer function:
Input sequence: Output sequence:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-36
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions:
Up to now we have found four possibilities to describe a linear, time invariant (LTI) system:
Impulse response, step response, frequency response, and transfer function.
For all of these four system representations we found also descriptions on how to generate the output sequence for a general input sequence. In the next few slides we will show how to transform the individual representations into each other. The results of these derivations are depicted on the next slide.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-37
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
Step response Impulse response Frequency response, transfer function
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-38
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
In order to obtain the transformation between the impulse and the step response of an LTI system we will have a closer look on the general signal representation (known from the previous slides): This signal representation must be also valid for and we obtain: Since we know that the step response is the system output for and the impulse response for , we get for the system output
… inserting the definition of the unit step function ...
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-39
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
If we finally simplify the summation index we obtain This means that the step response is a summation over the impulse response:
… substituting ...
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-40
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
Step response Frequency response, transfer function Impulse response
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-41
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
From our previous section we know that we can describe a unit impulse by a subtraction of two unit step sequences according to If we put both signals through an LTI system we obtain for the system outputs Thus we can conclude that the impulse response is equal to the difference of two shifted step responses.
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-42
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
Step response Frequency response, transfer function Impulse response
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-43
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
We know that in the time domain the output of an LTI system can be computed by means
In addition to that we know that a convolution corresponds in the Fourier and in the z domain to a multiplication: As a result we can see the relation between an impulse response and the corresponding frequency response and transfer function:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-44
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
Step response Frequency response, transfer function
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-45
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
In the part “Spectra” of this lecture we have derived the Fourier transform of a unit step
Before reusing that result we will exchange the last part (the last two terms):
… inserting the definition of cot(…) … … exchanging the cos- and sin-term … … excluding … … common denominator … … simplifying …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-46
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
As a result we get for the Fourier transform of the unit step function: By applying an inverse Fourier transform we obtain: Next we look at the output of an LTI system to such an input signal
… using that the step response is the system answer to a unit step and exploiting that an LTI system is a linear
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-47
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Output of the system to a unit step sequence (continued):
Relations between the system descriptions (continued):
… inserting that the response of LTI systems to is … … splitting the integral into two parts … … using that just one Dirac distribution is in the integration range …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-48
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Output of the system to a unit step sequence (continued):
Relations between the system descriptions (continued):
… using the sampling property of the Dirac distribution … … using that an integral over a Dirac distribution is 1 … … inserting the abbreviation of an inverse Fourier transform …
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Slide IV-49
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Our final result: This can be “rearranged” in order to find the inverse relation. However, before we have a look at this, we will first investigate the relation between the step response and the frequency response:
Relations between the system descriptions (continued):
… using that … … modifying such that we can use …
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Slide IV-50
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Inserting this result leads to Now we can resolve the equation for finding an inverse relation
Relations between the system descriptions (continued):
… bringing all terms that depend on to the left side … … applying a Fourier transform to both sides … … resolving for …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-51
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
During our studies on the z transform we found the following relation for summing a sequence and its impact in the z domain: This relation can be used to find the z transform of the step response. However, we start with the relation between a step and an impulse response
Relations between the system descriptions (continued):
… applying a z transform on both sides … … inserting the relation from above …
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Slide IV-52
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Z transform of the step response (continued): Resolving this equation for and applying an inverse z transform results in Now we have proved all necessary relations for our diagram!
Relations between the system descriptions (continued):
… using the abbreviation … … resolving for …
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Slide IV-53
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Relations between the system descriptions (continued):
Step response Frequency response, transfer function
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-54
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Partner work:
Partner work – Please think about the following questions and try to find answers (first group discussions, afterwards broad discussion in the whole group).
We tried to find now all relations among four different system descriptions. How
many relations are there in total? Please show a small diagram! …………………………………………………………………………………………………………………………………. …………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………….
Which relations do we miss? Please explain them!
…………………………………………………………………………………………………………………………………. …………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………….
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-55
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Reactions on elementary signals Reaction of LTI systems on exponential signals Reaction on general deterministic signals Relations between the system descriptions Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-56
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-57
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
A few slides before we have already defined the so-called BIBO stability (BIBO stands for bounded- input bounded-output). Our result was that a system is called BIBO stable if the output is limited according to if we can assure that the input is limited according to Now we will investigate stability conditions that can be derived from the parameters of a linear system. In order to keep it simple we will assume that the system has only one input and
BIBO stability:
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Slide IV-58
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
As a starting point we assume – as before – that we have a bounded input sequence: Now we try to derive an upper bound for the output sequence according to
BIBO stability (continued):
… inserting the definition of a convolution with a linear (but not necessarily time- invariant) system … … inserting the inequality of Cauchy and Schwarz … … inserting boundedness of the input sequence …
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Slide IV-59
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Result from the last slide: Obviously we can conclude that the output sequence is bounded if also the sum over the magnitudes of the coefficients of the impulse response is bounded: If the system is also time or shift invariant, we get for the stability condition:
BIBO stability (continued):
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Slide IV-60
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Some remarks:
The stability condition obtained before corresponds to the (sufficient) condition
for the existence of the Fourier transform
In contrast to the derivation when investigating spectra we have here also a
necessary condition: A linear system, that does not fulfill the condition is unstable!
From our previous investigations we know that a Fourier transform always exists for
a finite sequence. Thus, we can conclude that systems with a finite-length impulse response (so-called FIR systems) are always stable!
BIBO stability (continued):
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Slide IV-61
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
We know that the Frequency response is obtained by applying a Fourier transform to the impulse response . Thus, we obtain the same spectral symmetries for real-valued impulse responses that we get for real-valued signals: As a consequence we get even functions for the real part and for the magnitude of the frequency response and odd functions for the imaginary part and for the angle of the frequency response
Frequency response:
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Slide IV-62
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
The transfer function is computed by applying a z transform to the impulse response of linear, shift-invariant system: If we compute the transfer function for and for real-valued impulse responses we obtain:
Transfer function:
… for real-valued impulse responses we can use … … excluding the conjugate complex operation … … using the definition of the z transform …
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Slide IV-63
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
In the last slide we obtained for real-valued impulse responses Now we can conclude in addition:
If has a zero at , i.e. , than must have a zero at : If has a pole at , i.e. , than must have a pole
at : We can summarize: Poles and zeros of real systems appear as conjugate-complex pairs!
Transfer function (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-64
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-65
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes Probability density function Mean Auto and cross correlation Auto and cross power spectral density Variance Coherence
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Slide IV-66
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Assuming that we have a stochastic input sequence that is stationary and a linear system according to the following picture: Wanted are now the statistical properties of the system output. This sequence is also a stochastic process. We assume that can not be described directly, but by means of statistics, such as , ,
above – is also a stochastic process, we are looking now for the same statistical descriptions. Furthermore, we assume that the system can be described by, e.g., its impulse response or related
Introduction:
Linear system
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-67
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Unfortunately, the probability density function at the output of a linear system can not be described in a general manner. For the simple case of a memoryless system, which can be described by we obtain for the output probability density function This is obtained by applying a density transformation (see slides on the end of the chapter on „Discrete Signals and Random Processes”).
Probability density function:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-68
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
If the system output is generated by means of a difference equation according to and we can assume furthermore that samples of the input sequences with different index are statistically independent, we can obtain Here we used that summing independent processes results in convolving their densities. If the system is also shift invariant we can simplify the above equation to
Probability density function (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-69
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
In addition we can make use of the so-called central limit theorem. Several formulations of this theorem exist. One of them reads:
Assuming that are statistically independent random processes with arbitrary
probability density functions. For each process a (linear) mean value should exist and
and If all conditions from above are fulfilled the probability density function of the process converges with increasing to a Gaussian distribution with and .
Probability density function (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-70
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Probability density function (continued):
Central limit theorem: Matlab example for summing uniformly distributed random variables …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-71
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
First we assume only linearity (but not shift-invariance) of the system. In this case we can compute the system output as
Mean value:
… inserting the definition of a convolution … … exchanging the order of expectation and summation… … exploiting the stationarity of the input sequence and inserting the relation between impulse and step responses …
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Slide IV-72
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
If the system is shift invariant (in addition to linear) we can conclude furthermore: Thus, the transmission of the mean value is determined by frequency response at .
Mean value (continued):
… assuming shift invariance … … extending with 1 and using the definition of a Fourier transform …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-73
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Similar to the derivation of the mean value at the system output we obtain the auto and cross correlation sequence by applying the convolution. For the cross correlation sequence we get: After exchanging expectation and summation we obtain
Auto and cross correlation sequences:
… inserting the definition of a convolution … … inserting the abbreviated notation for the auto correlation …
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Slide IV-74
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
For stationary signals and shift-invariant systems the result of the last slide simplifies to This operation corresponds to a convolution. This means that we have the same relation between and as we have it for and .
Auto and cross correlation sequences (continued):
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Slide IV-75
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
In a similar manner we can compute the cross correlation between the system output and input of a linear and shift-invariant system (please note the different argument order in comparison to the last slides). We obtain: This cross correlation is created by convolution of the input auto correlation sequence with the mirrored conjugate complex impulse response .
Auto and cross correlation sequences (continued):
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Slide IV-76
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
If we would like to compute the auto correlation sequence at the output of a linear, shift- invariant system, we have to convolve the input auto correlation sequence (in arbitrary order) with and : If the input process is stationary we can simplify the equation to
Auto and cross correlation sequences (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-77
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Overview:
Auto and cross correlation sequences (continued):
Signals: Auto and cross correlation:
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Slide IV-78
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
The auto and the cross power spectral densities at the output of a linear, shift-invariant system are obtained by transforming the corresponding time-domain quantities: the auto or the cross correlation sequences. If convolutions are involved in the time-domain, multiplications with the corresponding frequency responses are resulting in the Fourier domain. Since convolutions with are involved, we will derive first the following relation:
Auto and cross power spectral densities:
… substituting and simplifying … … complex conjugation (twice) and using the definition of the Fourier transform …
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Slide IV-79
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
By transforming the results obtained in the time domain to the frequency domain we obtain the following relations:
Fourier transform z transform
Auto and cross power spectral densities (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-80
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Frequency domain relations (continued):
Discrete Fourier transform
Auto and cross power spectral densities (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-81
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
The squared magnitude of the frequency response is also called power transfer
sequence of an impulse response Corresponding quantities can be defined in the discrete Fourier domain and for Fourier series. Here, however, circular properties have to be considered.
Auto and cross power spectral densities (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-82
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Overview:
Auto and cross power spectral densities (continued):
Spectra: Power spectral densities:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-83
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Remarks:
The cross correlation sequence and the corresponding cross power spectral density
contain and , respectively, directly (beside the auto correlation sequence
For that reason the „cross“ analyses contain also information about the phase of a systen (in contrast to the „auto“ analyses, that contain only information about the magnitude of a system).
Practical hint: Measuring an impulse response can be performed by estimating the
cross correlation with white noise as excitation In comparison with direct measurements much lower signal amplitudes can be used.
Auto and cross power spectral densities (continued):
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-84
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Remarks (continued):
Overview about impulse response estimation by means of cross correlation analysis:
Auto and cross power spectral densities (continued):
Noise generator Linear system that should be identified Estimation of the cross correlation Should be fulfilled: Estimation of the variance of the excitation
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-85
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
If we know the auto correlation sequence and the mean of the system output we can easily compute the output variance (known from the slides of the section on “Discrete Signals and Random Processes”): Inserting the relations found before, we obtain Some more insight can be obtained by computing the terms in the Fourier domain. We can start with
Variance:
… inserting the definition of an inverse Fourier transform …
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Slide IV-86
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Inserting this intermediate result into our previous findings leads to Since we have , necessarily the integral must yield to a positive or at least to a non-negative value – even for “extremely narrow” . Therefore, the condition must hold for all frequencies!
Variance (continued):
… inserting our result for (see last slide) … Area under the input PSD weighted by the power transfer function!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-87
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
We start our studies on coherence with the following statement: Two processes are called “coherent”, if one of them can be described as the output of a linear, shift-invariant (LTI) system excited by the other one. The complex coherence function is defined as follows It can be shown that we get for the squared magnitude of this function
Coherence:
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-88
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
If we use the definition from the last slide for the following setup (assuming a linear, shift-invariant system with we obtain
Coherence (continued):
… inserting and …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-89
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Derivation (continued)
Coherence (continued):
… simplifying the square root in the denominator … … truncating the auto power spectral density … … only a phase term remains …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-90
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Derivation (continued) As a result we can conclude that the magnitude of the coherence between the input and the
However, this is only true if no distortions and no measurement noise is involved. If we extend our model according to we will obtain magnitude squared coherence values smaller than one.
Coherence (continued):
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Slide IV-91
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Coherence between a system input and a noisy system output: We assume now that the noise and the input signal are orthogonal. In this case all cross correlation sequences and all cross power spectral densities are zero:
Coherence (continued):
… inserting …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-92
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Inserting this assumption leads to: As a result we can conclude that now the magnitude of the coherence function is smaller than
Coherence (continued):
… inserting orthogonality between the input and the noise … … inserting and …
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-93
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Partner work:
Please ask your partner the questions on the extra slides and check her or his answers! Afterwards the questioning and the answering part will be exchanged!
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-94
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes Probability density function Mean Auto and cross correlation Auto and cross power spectral density Variance Coherence
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
Slide IV-95
Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems
This part:
System description System classification Discrete linear systems and their response to deterministic signals Stability of linear systems Special symmetries for real-valued systems Discrete linear systems and their responses to stochastic processes
Next part:
Idealized linear, shift-invariant systems