2006 Fall Welcome to Signals and Systems

2006 Fall Grading Policy Homework – 10% Lab – 20% Midterm Exam – 20% Final Exam – 50% Project

2006 Fall What will we learn? This course is about using mathematical techniques to help analyze and synthesize systems which process signals. We will learn an analytical framework: A language for describing signals and systems; A set of tools for analyzing signals and systems Problems of signal and system analysis: Analyzing existing systems; Designing systems.

2006 Fall Examples of signals Electrical signals --- voltages and currents in a circuit Acoustic signals --- audio or speech signals (analog or digital) Video signals --- intensity variations in an image (e.g. a CAT scan) Biological signals --- sequence of bases in a gene Economical signals ---price of stocks … we will treat noise as unwanted signals

2006 Fall Examples of systems Electrical systems --- amplifer circuit Computer systems --- mp3 player Control systems --- automobile Economical systems --- stock market

2006 Fall Signals and Systems Signals : functions of one or more independent variables. Systems : respond to particular signals by producing other signals or some desired behavior. As functions, they should have have define domain and range.

2006 Fall Signal Classification Types of Independent Variable Time is often the independent variable. Example: the electrical activity of the heart recorded with chest electrodes –– the electrocardiogram (ECG or EKG).

2006 Fall The variables can also be spatial Eg. Cervical MRI – In this example, the signal is the intensity as a function of the spatial variables x and y.

2006 Fall Independent Variable Dimensionality An independent variable can be 1-D (t in the EKG) or 2-D (x, y in an image). We focus on 1-D for mathematical simplicity but the results can be extended to 2-D or even higher dimensions. Also, we will use a generic time t for the independent variable,whether it is time or space.

2006 Fall Continuous-time (CT) Signals and Discrete-time (DT) Signals A continuous-time signal will contain a value for all real numbers along the time axis. X(t) A discrete-time signal will only have values at equally spaced intervals along the time axis. X[n] Why DT ? x(t)---sampling--->x[n] Can be processed by modern digital computers and digital signal processors (DSPs).

2006 Fall Signal Energy and Power Total Energy – Total energy over the time interval – Total energy over an infinite time interval Time-Averaged Power （平均功率） – Average power over the time interval – Average power over an infinite time interval Instantaneous power （瞬时功率）

2006 Fall Signal Energy and Power Energy signal 0 t Power signal 0 t Neither energy, nor power signal

2006 Fall Right- and Left-Sided Signals A right-sided signal is zero for t T, where T can be positive or negative.

2006 Fall Bounded and Unbounded Signals Whether the output signal of a system is bounded or unbounded determines the stability of the system.

2006 Fall Signal Operation Shifting y(t)=x(t-t 0 ) – t 0 >0 (delay) t 0 >0 – t 0 <0(advance)t 0 <0 Reflecting y(t)=x(-t) Reflecting Time-Scaling y(t)=x(at) – a>1(compress)a>1 – a<1(expand)a<1 It is rarely use the time-scaling operation when dealing with discrete waveform.

2006 Fall Signal Operation (1)Contract by a; (2) Shift right with – sign, shift left with + sign. (1)Reflect and contract by a; (2) Shift right with – sign, shift left with + sign.

2006 Fall Signals with symmetry Periodic signals – CT x(t) = x(t + T) – DT x[n] = x[n + N] If T1/T2=q/r, where q and r are integers, then there is T=rT1=qT2, which makes Demo: sum of periodic signals

2006 Fall Signals with symmetry (continued) Even and odd signals – Even x(t) = x(-t) or x[n] = x[-n] – Odd x(t) = -x(-t) or x[n] = - x[-n] x(0) = 0 or x[0] = 0 Any signals can be expressed as a sum of Even and Odd signals. That is:

2006 Fall Summary What we have learned in this lecture? – Examples and Classification of Signal – Signal operations – Signals with symmetry What was the most important point in the lecture? What was the muddiest point? What would you like to hear more about?

2006 Fall Readlist Signals and Systems – 1.3,1.4,2.5, Mathematical Review (P53) Question: – Periodic of DT Signals – Euler Relation

2006 Fall Problem Set 1.21(a),(c) 1.22(b),(d) 1.23(a) 1.24(b)