Slide 1 ELT2035 Signals & Systems Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Lesson 2 Introduction to systems Last lesson review ❑[.]
ELT2035 Signals & Systems Lesson 2: Introduction to systems Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Last lesson review ❑ Fundamental concepts: signals, data, information, systems ❑ Classification of signals CT/DT, Periodic/Nonperiodic, Causal/Anti-causal/Noncausal ❑ Energy and power of signals Energy signal and power signal ❑ Basic operations on signals Amplitude scaling, addition, multiplication, differentiation, integration Time scaling, reflection, time shifting ❑ Elementary signals Step, impulse, ramp, sinusoidal, real/complex exponential signals Fundamental concepts of systems ❑ Systems are used to process input signals in order to obtain the desired output signals ➢ A system may consists of physical components or may consists of an algorithm that computes the output signal from the input signal ➢ A system is characterized by its inputs, its outputs (or responses), and the rules of operation (or laws) that governs its behavior ❑ The mathematical description of the system’s rules of operation are called the model of the system ➢ Usually expressed as an operator H[.] that relates the outputs to the inputs, and commonly illustrated by the below “black box” concept Input signal x(t) System H[.] Output signal y(t) ❑ The study of systems includes major areas: mathematical modelling, system analysis, and system synthesis (design) ➢ Analysis: determine the output given the system and the input ➢ Synthesis/design: construct a system which will produce the desired output for the given input Systems as interconnections of operations ❑ The operator H[.] in the “black box” approach can also be viewed as a combination of basic operations performed on the input to yield the output ➢ Example: A 3-point discrete-time moving average system with input-output relationship 𝑦 𝑛 = 𝑥 𝑛 + 𝑥[𝑛 − 1] + 𝑥[𝑛 − 2] can be represented as follows ❑ In practice, systems are subject to unwanted signals that tend to disturb the operation of the system → noise Classification of systems ❑ Classification by configurations ➢ ➢ ➢ ➢ Single-input single-output (SISO) Single-input multiple-output (SIMO) Multiple-input single-input (MISO) Multiple-input multiple-output (MIMO) ❑ Classification by properties of the operator H[.] that represents the system ➢ ➢ ➢ ➢ ➢ ➢ ➢ Continuous time/discrete time Stable/unstable Instantaneous (memoryless)/dynamic (with memory) Causal/noncausal Invertible/non-invertible Time variant/time invariant Linear/non-linear ❑ A system may simultaneously possess several properties above, e.g Linear time invariant (LTI) systems Important properties of systems (1) ❑ Stability: a system is BIBO stable iff every bounded input results in a bounded output ❑ Memory: a system is said to possess memory if the output depends on past or future values of the input ➢ A system is memoryless if its output depends only on the current value of the input ❑ Causality: a system is causal if the present value of the output depends only on the present or past values of the input ➢ Noncausal system: the output depends on at least one future value of the input → NOT capable of operating in real time ❑ Invertibility: a system is invertible if the input can be recovered from the output ➢ The process of inverting the output is characterized by the operator Hinv such that HinvH = I (identity operator) The system associated with Hinv is called the inverse system (for example: network equalizer) Important properties of systems (2) ❑ Time invariance: a system is time invariant if a time shift in the input leads to an identical time shift in the output ➢ The characteristics/behaviours of a time invariant system not change with time ❑ Linearity: a system is said to be linear in terms of the system input 𝑥(𝑡) and the system output 𝑦(𝑡) if it satisfies the following two properties ➢ Superposition: if the system respectively produces outputs 𝑦1 (𝑡) and 𝑦2 (𝑡) to the input 𝑥1 (𝑡) and 𝑥2 (𝑡), then the composite input 𝑥1 𝑡 + 𝑥2 (𝑡) must yield the corresponding output 𝑦1 𝑡 + 𝑦2 (𝑡) ➢ Homogeneity: whenever the input 𝑥(𝑡) is scaled by a factor 𝑎, the output 𝑦(𝑡) must be scaled by the same constant factor 𝑎 ❑ In this course, we will only work with linear time invariant (LTI) systems Exercise #1 Consider x(t) defined by x(t) = 0.5t, for ≤ t ≤ and x(t) = 0, for t < and t ≥ Plot x(t) for t in [0, 5] Is x(t) a signal? If not, modify the definition to make it a signal Exercise #2 A triangular wave is depicted in the below figure Is it periodic or aperiodic? Periodic What is its fundamental frequency? 1/0.2=5Hz Is it an energy signal or a power signal? Power What is the average power of this signal? 1/3 Exercise #3 Exercise #4 Prove that every signal 𝑓 𝑡 can be expressed as a sum of even and odd signals Find the even and odd components of 𝑓 𝑡 = 𝑒 𝑗𝑡 Solution: Rewrite 𝑓 𝑡 as 𝑓 𝑡 = 𝑓 𝑡 + 𝑓(−𝑡) + 𝑒𝑣𝑒𝑛 𝑓 𝑡 + 𝑓(−𝑡) 𝑜𝑑𝑑 Applying previous results for 𝑓 𝑡 = 𝑒 𝑗𝑡 , we obtain 𝑗𝑡 𝑓𝑒 𝑡 = 𝑒 + 𝑒 −𝑗𝑡 = cos(𝑡) 𝑗𝑡 𝑓𝑜 𝑡 = 𝑒 − 𝑒 −𝑗𝑡 = 𝑗 sin(𝑡) Exercise #5 Consider the rectangular pulse 𝑥 𝑡 of unit amplitude and a duration of units Sketch 𝑦 𝑡 = 𝑥(2𝑡 + 3) Exercise #6 ∞ Show that −∞ 𝑒 −2 𝑥−𝑡 𝛿 − 𝑡 𝑑𝑡 = 𝑒 −2 𝑥−2 Solution: Applying the sampling property of the Dirac function: 𝑒 −2 𝑥−𝑡 𝛿 − 𝑡 = 𝑒 −2 𝑥−𝑡 ห2−𝑡=0 𝛿 − 𝑡 = 𝑒 −2 𝑥−2 𝛿 − 𝑡 ∞ Hence −∞ 𝑒 −2 ∞ 𝑥−𝑡 ∞ 𝛿 − 𝑡 𝑑𝑡 = −∞ 𝑒 −2 𝑒 −2 𝑥−2 −∞ 𝛿 − 𝑡 𝑑𝑡 = 𝑒 −2 Dirac function 𝑥−2 𝑥−2 𝛿 − 𝑡 𝑑𝑡 = according to the definition of Exercise #7 a Consider a CT system whose input x(t) and output y(t) are 𝑡+1 related by 𝑦 𝑡 = =𝜏0 𝑥 𝜏 𝑑𝜏 for 𝑡 > Is the system memoryless? stable? causal? b Consider a DT system whose input and output are related by 𝑦 𝑛 = 3𝑥 𝑛 − − 0.5𝑥 𝑛 + 𝑥 𝑛 + Is the system memoryless? stable? causal? c Consider 𝑦 𝑡 = cos 𝜔𝑐 𝑡 𝑥(𝑡) Is the system memoryless? linear? time-invariant? d Consider 𝑦 𝑛 = 2𝑛 + 𝑥 𝑛 Is the system memoryless? linear? time-invariant? Solution: a Dynamic, stable, noncausal b Dynamic, stable, noncausal c Memoryless, linear, time varying d Memoryless, linear, time varying Exercise #8 The output of a discrete-time system is related to its input as follows 𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑥 𝑛 − + 𝑎2𝑥 𝑛 − + 𝑎3𝑥 𝑛 − Let the operator 𝑆 𝑘 denote a system that shifts 𝑥 𝑛 by 𝑘 time units to produce 𝑥[𝑛 − 𝑘] Find the operator H for the system relating 𝑦 𝑛 to 𝑥 𝑛 Sketch the block diagram for H Solution: Using operator 𝑆 𝑘 , we can rewrite 𝑦 𝑛 as 𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑆1 𝑥 𝑛 + 𝑎2𝑆 𝑥 𝑛 + 𝑎3𝑆 𝑥 𝑛 = (𝑎0 + 𝑎1 𝑆1 + 𝑎2 𝑆 + 𝑎3 𝑆 ) 𝑥 𝑛 = 𝐻 𝑥 𝑛 Thus 𝐻 = 𝑎0 + 𝑎1 𝑆1 + 𝑎2 𝑆 + 𝑎3 𝑆 See the next figure Exercise #9 Consider the DT system given in exercise #8: 𝑦 𝑛 = 𝑎0𝑥 𝑛 + 𝑎1𝑥 𝑛 − + 𝑎2𝑥 𝑛 − + 𝑎3𝑥 𝑛 − with constant coefficients 𝑎0 , ⋯ , 𝑎3 are finite Is system BIBO stable? Why? Solution: Using the given input-output relation 𝑦 𝑛 = 𝑎0 𝑥 𝑛 + 𝑎1 𝑥[𝑛 − 1] + 𝑎2 𝑥 𝑛 − + 𝑎3 𝑥 𝑛 − we may write 𝑦 𝑛 = 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛 − + 𝑎2 𝑥 𝑛 − + 𝑎3 𝑥 𝑛 − ≤ 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛 − + 𝑎2 𝑥 𝑛 − + 𝑎3 𝑥 𝑛 − ≤ 𝑎0 𝑀𝑥 + 𝑎1 𝑀𝑥 + 𝑎2 𝑀𝑥 + 𝑎3 𝑀𝑥 , where 𝑀𝑥 = 𝑥 𝑛 Hence, provided that 𝑀𝑥 is finite, the absolute value of the output will always be finite It follows therefore that the system is BIBO stable