COSC 3451: Signals and Systems Instructor: Dr. Amir Asif Department of Computer Science York University Handout # 2 Topics: 1. CT and DT Systems 2. Parallel and Cascaded Connections 3. Properties of Systems: Memory, Causality, Stability, linearity 4. Linear and Time Invariant Systems 5. Convolution Integral and Sum Oppenheim: Section 1.5 – 1.7, 2.1 – 2.2

Systems System: processes an input signal producing an output signal with certain features of the input signal enhanced and other features unchanged or diminished. Examples of Systems: Fig. 1: RLC Series Circuit Fig. 2: Spring-mass-damper mechanical system

Classification of Systems Continuous-time System: is one in which a continuous-time input results in a continuous-time output. Figures 1 and 2 are both CT systems represented by constant coefficient linear differential equations. Discrete-time System: is one in which a discrete-time input results in a discrete-time output Example is your bank account where the monthly balance y[n] is incremented by the additional deposit x[n] and interest at a rate of r% per month. DT systems are represented by linear constant coefficient finite difference equations.

Interconnection of Systems Cascaded (Series) Configuration: Parallel Configuration: Feedback (Series-Parallel) Configuration:

Properties of Systems (1) Memoryless Systems: in which the output at a particular instant of the independent variable is only dependent upon the value of input at the same instant. Otherwise, the system has a memory Activity 1: Which of the following systems are memoryless? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal:

Properties of Systems (2) Invertible Systems: If an output of a system can be processed by another system to produce the original input, the system is said to be invertible. Basic condition of invertibility: Distinct inputs lead to distinct outputs. Activity 2: Which of the following systems are invertible? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal:

Properties of Systems (3) Causal (Non-anticipative) Systems: in which the output signal at a particular instant of the independent variable depends upon the value of input signal either in the past or at the same instant. Otherwise, the system is called a noncausal (or anticipative) system. Activity 3: Which of the following systems are causal? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal:

Properties of Systems (4) Stable Systems: are those which produce a bounded output for every bounded input. The above condition of stability is called BIBO stability. Activity 4: Which of the following systems are stable? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal: e. Miscellaneous:

Properties of Systems (5) Linear Systems: satisfy the principle of superposition: If an input consists of a weighted sum of several inputs (for which individual responses) are known, then the output is the same weighted sum of the individual responses. Activity 5: Which of the following systems are stable? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal: e. Miscellaneous:

Properties of Systems (6) Time Invariant Systems: If an input is shifted in time then the output gets shifted by a similar shift without any other change. Activity 6: Which of the following systems are time invariant? a. Amplifier: b. Accumulator: c. Decimator: d. Time Reversal: e. Miscellaneous:

Linear Time Invariant Systems (1) Linear Time Invariant (LTI) Systems: are those that satisfy the linearity and time-invariant properties. LTI systems are special because these are completely represented by constant coefficient linear differential equations for CT systems or constant coefficient linear difference equation for DT systems. LTI systems are also completely represented by its impulse response. Bullets 3 and 4 imply that given the representative impulse response (or equivalent differential or difference equation), the output to any input can be computed. Impulse Response of LTI Systems

LTI Systems: Convolution Sum For DT system, the output is the convolution sum of the input and the impulse (unit sample response) Activity 7: Calculate the output if the input and the impulse response are given by

LTI Systems: Convolution Integral For CT system, the output is the convolution integral of the input and the impulse (unit sample response) Activity 8: Calculate the output if the input and the impulse response are given by