Week 1 The Signals and Systems Abstraction The system transforms an input signal into an output signal Both the input and output are signals A signal is a mathematical function with an independent variable i e time and dependent variable The system is described by the way that it transforms the input signal into the output signal All useful models focus on the most relevant signals and ignore those of lesser signi cance Check yourself 1 List at least four possible output signals for the car steering problem The car s three dimensional position The car s angular position The rotational speed of the wheels The temperature of the tires Modularity primitives and composition In a composite system the steering controller determines t which is input into the car The car generates po t which is subtracted from pi t to get e t which is input to the steering controller The triangular component is called a gain or scale of 1 Its output is equal to 1 times its input The triangle symbol is used to indicate that we are multiplying all the values of the signal by a numerical constant which is shown inside the triangle The dashed red box illustrates modularity of the signals and systems abstraction Discrete Time Signals and Systems Signals whose independent variables are discrete i e take only integer values Some signals are found in nature For example the primary structure of DNA is described by a sequence of base pairs Discrete time signals are found in computers For example the difference between the desired position pi t and our actual position po t is an error signal e T which is a function of continuous time t If the controller only observes this signal at regular sampling intervals T then its input could be regarded as a sequence of values x n that is indexed by the integer The relationship between the discrete time sequence x n and the continuous signal x t is given by x n x nT where T is the length of one time step Sampling converts a signal of continuous domain to one of discrete domain For example images are typically represented as arrays of pixels accessed by integer valued rows and columns rather than as continuous brightness elds indexed by real valued spatial coordinates Linear Time Invariant Systems State machines allow us to specify any discrete time system whose output is computable from it history of previous inputs The representation of systems as state machines allows us to execute a machine on any input we would like in order to see what happens Execution lets us examine the behavior of the system for any particular input for any particular nite amount of time but does not let us characterize any general properties of the system of its long term behavior A small but powerful subclass of the whole class of state machines called discrete time linear time invariant LTI systems which will allow deeper forms of analysis In LTI systems Inputs and outputs are real numbers The state is some xed number of previous inputs to the system as well as a xed number of previous outputs of the system The output is a xed linear function of the current input and any of the elements in the state We will restrict our attention to the case where the input is a single real number and the output is a single real number LTI systems can be analyzed mathematically in a way that lets us characterize some properties of their output signal for any possible input signal Another important property of LTI systems is that they are compositional the cascade parallel and feedback combinations of LTI systems are themselves LTI systems Discrete Time Signals The PCAP system for discrete time signals A signal is an in nite sequence of sample values at discrete time steps A capital X stands for the whole input signal and x n stands for the value of signal X at time step n If there is a single system under discussion use X for the inpiut signal to that system and Y for the output signal We wil say that systems transduce input signals into output signals Unit Sample Signal We will work with a single primitive called the unit sample signal It is de ned on all positive and negative integer indices as follows n Our rst operation will be scaling or multipication by a scalar The result of multiplying any signal X by a scalar c is a signal so that if Y c X then y n c x n Scaling Delay The next operation is the delay operation The result of delaying a singal X by a new signal RX such that if Y RX then y n x n 1 Addition of signals is accomplished component wise so that if Y X1 x2 then y n x1 n x2 n Addition Finally we can add two signals together Advancing skipped Algebraic Properties of Operation and Signals Adding is commutative and associative Scaling is commutative Scaling distributes over addition Furthermore R distributers over addition and scaling These algebraic relationships mean that we can rewrite 3 4R 2R 2 as 3 4R 2R 2 Feedforward Systems Representing Systems Operator equation Difference Equation Block Diagrams A subclass of discrete time LTI systems which are exactly those that can be described as performing some combination of scaling delay and addition operations on the input signal We can represent systems using operator equations difference equation block diagrams and Python state machines An operator equation is a description of how signal are related to one another using the operations of scaling delay and addition on whole signals Consider a system that has an input signal X and whose output signal is X XR We can describe the system using the operator equation Y X XR We cab rewrite this as Y 1 R X Feedforward systems can always be described using an operator equation of the form Y X where is a polynomial in R An alternative representation of the relationship between signals is a difference equation A difference equation describes a relationship that holds among samples values at particular times of signals The operator equation Y X RX can be expressed as this equivalent equation y n x n x n 1 The operation delaying a signal can be seen here as referring to a sample of that signal at time step n 1 Difference equations are convenient for step by step analysis letting us compute the value of an output signal at any time step given the values of the input signal Another way of describing a system is by drawing a block diagram which is made up of components connected by lines with arrow on them These are primitive components corresponding to our operations on signals These lines represent signals All
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