SIG: Signal-Processing

Transcription

1 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz SIG: Signal-Processing Copendiu (6) Prof. Dr.-Ing. Rainer Bartz rainer.bartz@th-koeln.de Contact: eail: website: office: rainer.bartz@th-koeln.de (contains course aterial according to seester progress) ZW6- Recoended Textbooks: Carlson, Gordon E.: Signal and Linear Syste Analysis; John Wiley & Sons (SIG only covers the discrete-tie parts of the book; chapter references are given by "(Cx.y)") Sith, Steven W.: The Scientist and Engineer's Guide to Digital Signal Processing; California Technical Publishing (SIG goes beyond the contents of this book in soe aspects, but does not cover all of the book; free download of pdf-chapters at Oppenhei, Alan V.; Schafer, Roland W.: Discrete-Tie Signal Processing; Prentice Hall (SIG only covers parts of this book; Geran book version is available) and any ore SIG_Copendiu_4.doc /.7.6 / RBz a

2 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz SIGNALS AND SYSTEMS. Definitions (C.) Syste := part of the real world that ay be enclosed by a (real or virtual) envelope/boundary. Signal := entity that carries inforation A syste ay have input, output, and internal signals. An input (output) signal carries inforation across the syste boundary into (out of) a syste. Often signals are functions of the tie t.. Matheatical Models (C.) A quantitative description is desirable to understand signals and systes. This is in ost cases only possible to a certain degree of accuracy, thereby obtaining a odel of the signal resp. syste. Matheatical ethods are used: A syste is odeled by a set of equations. A signal is odeled by a function expression. Benefit of such odels: they ay be further analyzed/used with atheatical eans. Drawback of odels: they do not exactly fit to reality. Trade-off: The better they shall eet reality the ore coplex they becoe. SIG works on odels (without always explicitly noting this). Engineering deals with following tasks: odeling a syste: find odels that are accurate enough analyzing a syste: observe the behavior of the syste (ostly by regarding its odel) designing a syste: deterine/adapt structure and/or paraeters of a syste so that a desired behavior is produced. SIG focuses on the analysis of systes Continuous-Tie (CT) vs. Discrete-Tie (DT) (C.3) CT-signal: holds inforation at any given point in tie. DT-signal: holds inforation only at distinct points in tie; each value at such distinct point is called a saple; distinct points in tie ay be equally spaced (equidistant saples). A DT-signal is produced when a CT-signal is sapled. CT-syste: a syste whose odel only contains CT input, output, and internal signals. DT-syste: a syste whose odel only contains DT input, output, and internal signals. Hybrid syste: a syste whose odel ixes CT and DT signals. SIG focuses on equidistant DT-signals and DT-systes. Notation: CT-signal: x(t). Value at t=t: x(t). DT-signal (equidistant saples): x[nt], x[n], xn; SIG will not use the latter. T := saple spacing: tie distance between adjacent distinct points; is constant in SIG. fs := saple rate; fs = /T n := saple index; ; ordinal nuber of the saple; saple at t= has n=; saples at t> have n>. Value at t=nt: x[nt], x[n]. Reark to physical units: SIG usually oits units, assuing that all nubers given are ultiples of the corresponding SI unit..4 Sapling Theore of Shannon and Nyquist Given a CT-signal x(t). Let fax be the largest frequency contained in x(t). If x(t) is sapled with a rate fs >. fax, resulting in a DT signal x[nt], none of the inforation of x(t) is lost by such sapling, and x(t) can copletely be reconstructed fro x[nt].. fax is called the Nyquist frequency of x(t). If this theore is not obeyed, aliasing occurs. SIG_Copendiu_4.doc /.7.6 / RBz a

3 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz BASICS ON SIGNALS AND SYSTEMS. Tie-related Operations on DT-Signals (C.).. Tie Reversal of DT-Signals A DT-signal x[nt], when tie-reversed, results in a different (but siilar) signal y[nt] = x[nt]. Graphically: y[nt] appears to be x[nt], but irrored at the location t= (i.e. n=). Matheatically: Tie reversal is achieved by replacing 'n' by 'n' at each occurrence of 'n' in the expression defining x[nt]... Tie Shift of DT-Signals A DT-signal x[nt], when tie-shifted by a value nt, results in a different (but siilar) signal y[nt] = x[(n- n)t]. Graphically: When n>, y[nt] appears to be x[nt], but shifted by nt towards positive tie axis (to the right); when n<, y[nt] appears to be x[nt] but shifted by nt towards negative tie axis (to the left). Matheatically: Tie shift is achieved by replacing 'n' by 'n-n' at each occurrence of 'n' in the expression defining x[nt]. Note that a tie shift ust always be a ultiple of T...3 Cobined Operations These operations ay be concatenated (perfored subsequently in any cobination). Note however that the sequence atters as soon as tie shift and tie reversal are both involved.. Characteristics of Signals (C.).. deterinistic <-> stochastic Deterinistic signals have a tie trace that is known for all points in tie both in the past as well as in the future. The values of stochastic signals for each point in tie are not known a-priori; however soe rules apply (e.g. on the range and distribution of the values). SIG does not cover stochastic signals siply defined <-> piecewise defined Siply defined signals ay be specified by one atheatical expression for the whole range of n. Piecewise defined signals divide the range of n into intervals and specify one atheatical expression for each interval. Building sus (discrete 'integration') over subsequent saples ust occur individually for each interval, adding these results afterwards; also building differences (discrete 'derivation') ust occur individually in each interval. Soe frequently used types of piecewise defined signals have been given shortcut notations (e.g... )...3 even signal <-> odd signal An even signal is syetric to the vertical axis at t=; it is identical to its tie reversed counterpart: x[nt] = x[nt]. An odd signal is syetric to the point ( ): x[nt] = x[nt]. Most real-world signals are neither even nor odd. Any x[nt] ay be coposed of the su of an even signal xe[nt]and an odd signal xo[nt] as x[nt]= xe[nt] + xo[nt] with xe[nt] = ( x[nt] + x[nt] ) / xo[nt] = ( x[nt] x[nt] ) /..4 quantized A quantized signal will show only saple values that are an integer ultiple of a quantu (the sallest positive value that can appear). Only distinct values will thus appear in vertical axis direction. This is usually the case when representing a signal as a sequence of nubers within a coputer syste, and special care should be given to the accuracy and value range in those cases. Moreover, besides the quantu another iportant issue in floating point nubers is the resolution. Due to the restricted tie available, SIG cannot cover the specific issues involved with quantized signals; if quantization effects are expected, the corresponding literature should be consulted...5 periodic signal <-> aperiodic signal A signal is periodic if (and only if) a signal-section of length n. T > can be identified that repeats infinitely along the tie axis. T = n. T is called period of the signal. Matheatically defined: A DT-signal is periodic if (and only if) a value n> exists so that x[(n-n)t] = x[nt]. SIG_Copendiu_4.doc /.7.6 / RBz 3a

4 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz Note that any ultiple of T is also a period. The sallest available T that fulfills the condition is forally called 'fundaental period'; however in any cases 'period' stands for that sallest available value. Note that sapling a periodic CT-signal x(t) does not necessarily result in a periodic DT-signal x[nt], and even if x[nt] is periodic, its period ay differ fro the period of x(t)...6 borders and extent Intuitively the left border of a signal is where it starts and the right border is where it ends. Its extent is the width of the signal. Matheatically defined: The left border of a signal x[nt] is the sallest index n (resp. sallest point in tie nt) where x[nt]. The right border of a signal x[nt] is the largest index n (resp. largest point in tie nt) where x[nt]. The extent of a signal x[nt] is the nuber of saples ne (resp. tie width te) of the sallest interval containing all saples x[nt]. If the extent is finite, the signal ay be specified by an ordered set of values (plus indicating the value at t=)..3 Representation of DT-Systes (C.3).3. Graphical Representations Though a DT-syste ay be graphically represented by a coponent diagra, showing the physical coponents it is coposed of, a ore widespread representation of DT-systes is the block diagra. A block diagra is a ore abstract and generalized representation. It consists of any nuber of the following eleents: () block: a block ay have a nuber of input and output signals, though typically a block diagra is preferred where blocks have one input and one output (SISO: single input single output). It uses the input signal(s) to produce the output signal(s). The function it perfors is often denoted within the graphical icon of the block. () connection: a connection is a directed line; it represents a signal. (3) cobination: a cobination ay atheatically build an output signal fro a set of input signals applying basic functionality like add or ultiply. (4) branch: a branch picks the signal at a specific location in the block diagra for use at another location. It does not odify the signal it is branched off. Note that such diagras are not necessarily unique for a syste; the sae syste ay be represented by a variety of diagras Matheatical Representations A syste ay be represented by a set of equations. Such equations ay include input, output, and internal signals. As with graphical representations, such set of equations need not be unique for a syste. A syste ay finally be represented by a set of independent syste equations. These are special equations that obey to the following rules: () A syste equation ay not contain any internal signal. () A syste equation ay only contain one output signal; besides this, it ay contain any nuber of input signals. (3) The nuber of output signals of a syste deterines the nuber of independent syste equations. They are sufficient to uniquely specify the syste. (This iplies that SISO systes are uniquely described by one syste equation.) In ost cases it is desirable to find a (set of) syste equation(s) before further analyzing a syste. Syste equations of CT-systes generally are differential equations; they are not further investigated in SIG. Syste equations of DT-systes generally are difference equations. Note the slight (but iportant) naing variation..4 Characteristics of DT-Systes (C.4) (this and all subsequent sections are assuing SISO systes).4. syste with eory <-> eoryless syste In a eoryless syste the output signals at a given point in tie depend only on the input signals at the sae point in tie. In a syste with eory the output signals at a given point in tie depend also on signals at other points in tie..4. causal syste <-> non-causal syste A syste is causal if its output signals at a given point in tie nt depend only on the input signals at ties nt nt. A syste is non-causal if its output signals at a given point in tie nt depend also on the input signals at any tie nt > nt. All real-world systes are causal. Each eoryless syste is causal. SIG_Copendiu_4.doc /.7.6 / RBz 4a

5 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz order of a syste The order of a syste is a positive integer nuber that gives an ipression of the syste's coplexity. It ay usually be read as the highest appearing index-shift in the syste equation..4.4 linear syste <-> nonlinear syste Intuitively a syste can be identified as linear, if () putting a ultiple of an input signal to the syste input results in ultiplying the output signal with the sae factor, and () putting a su of several signals to the syste input results in an output which is built fro the su of the output signals that each of the individual input signals would produce. Matheatically defined: If a syste produces () an output y[nt] as result of an input x[nt] and () an output y[nt] as result of an input x[nt], and if then a cobined input signal x[nt] = a x[nt] + b x[nt] results in an output signal y[nt] = a y[nt] + b y[nt] for any such signals x[nt] and x[nt] and any paraeters a,b, the syste is said to be linear. Most real-world systes are nonlinear; however in any cases it is applicable to treat the as nearly linear around an operating point. This however is not further discussed in SIG; it will assue linear systes without investigating such linearization processes..4.5 tie-variant syste <-> tie-invariant syste A syste is tie-invariant if a tie shift of input signals leads to exactly the sae tie-shift at the output signals (without any other effects). This finally eans that the syste behavior does not change over tie; its paraeters are constant. Matheatically defined: If a syste produces an output y[nt] as result of an input x[nt], and if then a tie-shifted input signal x[(n-n)t] results in an output signal y[(n-n)t] for any signal x[nt] and any tie shift nt, the syste is said to be tie-invariant Most real-world systes are tie-variant, but ay be considered tie-invariant if the observation interval is significantly shorter than the tie it takes for syste paraeters to change. Systes that are tie-invariant and linear are called LTI-systes. They build up a very iportant group of systes (resp. syste odels) as a quite coplete atheatical theory exists to work with the..4.6 stability of a syste Different definitions of stability exist. Of the the BIBO(bounded input bounded output)-stability shall be considered here. A syste is stable in this sense if it reacts on any bounded input signal with a bounded output signal. This definition however is not very helpful in deterining stability of a particular syste. See future chapters on this topic. SIG_Copendiu_4.doc /.7.6 / RBz 5a

6 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz DTSIGNALS 3. Sinusoidal Signals (C.) cosine-signal: x[nt] = A cos(fnt + ) ;cosine-representation is preferred sine-signal: x[nt] = A sin(fnt + s) Independent paraeters: A : aplitude; A>; negative values ust be ade positive by a phase shift of. f : frequency T = /f : period; = f : angular frequency : phase (angle) s = + / : sine phase angle Note that a DT sine-signal can only be represented by a DT cosine-signal (and vice versa) if T/4 is an integer ultiple of T. Note further that a DT cosine- or sine-signal is not necessarily periodic, and if it is, its period ay differ fro T. Relevance: An LTI syste will not change the frequency of a sinusoidal signal. It reacts on a sinusoidal signal at its input with a sinusoidal signal of the sae frequency at its output; only aplitude and phase ay be changed. Thus for LTI systes it is sufficient to just deterine the syste's influence on aplitude and phase. 3. Exponential Signals (C.) exponential signal: exponentially daped sinusoidal signal: n n x[nt] nt for ; deterines the steepness of rise (<) or fall (>). X e n n x[nt] X e nt cos(f nt ) for n n n n Relevance: Energy-transport echaniss in nature behave according to these functions in any cases. ;X e nt is the envelope Singularity Signals (C.3) pulse function: x[nt] A [nt] A: aplitude, weight of the pulse unit pulse function: n [nt] for n (unit) step function: n u[nt] for n (alternative sybols: [nt] in telecounications, [nt] in controls) Note: to not confuse the step function for a voltage (which in Gerany is also using u as notation), SIG will always subscript the sybol u in case a voltage is eant (e.g. u[nt]). rap function: x[nt] = A. r[(n-n)t] A gives the slope of the rap with regard to the tie axis, nt its start on the tie axis unit rap function: n r[nt] for nt n It ay also be represented as r[nt] = nt. u[nt]. Relevance of singularity signals: These signals are very useful in the analysis of ideal systes. [nt] ay denote a singular event, and any signal with finite extent can be represented as a finite su of weighted pulses. Moreover, the unit pulse response h[nt] of an LTI-syste contains all knowledge about the syste. u[nt] ay be used to describe switching or tie-liited processes. Also, any piecewise linear signal can be represented as a su of step and rap functions. 3.4 Periodic Signals Soe function expressions for signals are inherently periodic (e.g. cos(n/5) ). However not all periodic signals have such siple atheatical representation. Two ain approaches can be found to define the: 3.4. periodic repetition of one period The atheatical specification of a periodic signal x[nt] with period T=nT often uses a two-step approach: SIG_Copendiu_4.doc /.7.6 / RBz 6a

7 TH Köln - Technology, Arts, Sciences Prof. Dr. Rainer Bartz () the signal xp[nt] of just one (arbitrary) period is specified as xp[nt] = {... () the periodic signal x[nt] is defined as an infinite su of tie-shifted xp[nt] signals with tie-shift being an integer ultiple of nt: x[nt] xp i [(n in )T] Thus a periodic signal is uniquely defined by xp[nt] and n superposition of a set of orthogonal periodic basis functions; Fourier series Given an arbitrary periodic signal x[nt] with period T=nT. It can be shown that the signal x[nt] is exactly represented by the su x[nt] n X e j f nt with f=/t. The su is called the Fourier series of x[nt], and the (constant and usually coplex) values X are called the Fourier coefficients. They can be deterined by X n n nn n x[nt] e - j f nt ;[,,..,n-]; for arbitrary n f is called the fundaental frequency;. f is the th haronic frequency. It can be shown that the sequence of X values repeats periodically when extending beyond the interval [,n-], including <. X ay have aplitude and phase, which provide inforation about aplitude and phase of a cosine contribution at frequency. fo. For real signals x[nt] the aplitude is even and the phase is odd. The set of X values can graphically be presented as a function of the frequency; they are called the spectru of x[nt]. This usually requires two diagras, one for the aplitude values X and one for the phase values X. Alternative representations of the Fourier series: n n j f nt j X j f nt coplex exponential series: x[nt] X e X e e cosine series: x[nt] X n / X cos(f nt X ) (preferred) n / n / a a X cos( X ) trigonoetric series: x[nt] a cos(f nt) b sin(f nt) with b X sin( X ) For even n the su ters at =n/ (if any) should be zero; if not, adding half of the to cosine and trigonoetric series ay still recover x. 3.5 Signal Energy and Signal Power (C.4) signal energy: N E x li T x[nt] ; Ex T x[nt] N n-n n- signal power: N P x li x[nt] N N n-n RMS(root ean square): RMSx Px Signal classification: energy signal: a signal whose signal energy Ex is <Ex<; its signal power then is Px=. power signal: a signal whose signal power Px is <Px<; its signal energy then is Ex=. other signals (either Ex= or Px=) Note: signal energy is different fro physical energy, signal power is different fro physical power. Those physical quantities describe the effects at real coponents while signal attributes exist independent fro coponents. Note: periodic signals typically are power signals. The signal power Px of a periodic signal x[nt] with period T=nT can be deterined either fro the forula above, n n - or fro P x x[nt] n nn (for any n); just suing over one period; this is the ean value of x[nt]. using whatever is easier to evaluate. SIG_Copendiu_4.doc /.7.6 / RBz 7a