Introduction

1. Introduction Chapter 1

2. Signals • A signal is a function of time, e.g., • f is the force on some mass • vout is the output voltage of some circuit • p is the acoustic pressure at some point • notation: • f, vout, p or f(.), vout(.), p(.) refer to the whole signal or function • f(t), vout(1.2), p(t + 2) refer to the value of the signals at times t, 1.2, and t + 2, respectively • for times we usually use symbols like t, t , t1, . . .

3. Signal Example

4. Real Signals • AM radio signal • FM radio signal • cable TV signal • audio signal • NTSC video signal • 10BT Ethernet signal • telephone signal

5. System • a system transforms input signals into output signals • a system is a function mapping input signals into output signals • we concentrate on systems with one input and one output signal, i.e., single-input, single-output (SISO) systems • notation: • y = S(u) means the system S acts on input signal u to produce output signal y

6. Block System • systems often denoted by block diagram • boxes denote systems; arrows show inputs & outputs • lines with arrows denote signals (not wires) • special symbols for some systems

7. System Example

8. Signals and Systems • Modeling the physical world • Physical system (e.g., LRC circuit) – using mathematical equation • Input/output signal – using mathematical function

9. Signals and Systems • Example: LRC • LRC represented by a mathematical Equation • ordinary diff. eqn. • No sampling (continuous time system) • V(i) is a mathematical function

10. Signals and Systems - Examples Different systems can be MODELED using the same mathematical function

11. Signals and Systems - Examples Human speech production system — anatomy and block diagram

12. Signals and System Categorizations • Continuous time (analog) • Discrete time (digital)

13. Systems Described in Differential Equations • Many systems are described by a linear constant coefficient ordinary differential equation (LCCODE)

14. Second Order Continuous System • Second-order RC circuit • Closed loop system Find the mathematical relationship in terms of input & output • Remember: • v1-y = iR2 • v1=iR2+y and i(t) =C dv/dt Substitute: The 2nd order diff eqn can be solved using characteristic equation or auxiliary equation

15. Continuous System Example • A digital player/recorder Processor Analog/Digital Converter Digital/Analog Converter Reconstructed Digital Signal Sampling Signal Digital Output Analog Input

16. Sample Matlab Code To Generate Signal on the Soundcard! • %%%%%%% • % The following program will send a 500 Hz sine wave to analog • % output channel 1 for one second. • %%%%%%% • %%Open the analog device and channels • AO = analogoutput('winsound',0); • chan = addchannel(AO,1); • %% Set the sample rate and how long we will send data for • %% 44,100 Hz, 1 seconds of data • duration = 1; %in seconds • frequency = 500 %in Hz • SampleRate = 44100; • set(AO,'SampleRate',SampleRate) • set(AO,'TriggerType','Manual') • NumSamples = SampleRate*duration; • %% Create a signal that we would like to send, 500 Hz sin wave • x = linspace(0,2*pi*frequency,NumSamples); • y = tan(sin(1*x))' - sin(tan(1*x))'; • %y = sin(x)'; • %data = y • data = awgn(y,10,'measured'); % wite noise • %% Put the data in the buffer, start the device, and trigger • putdata(AO,data) • start(AO) • trigger(AO) • %% clean up, close down • waittilstop(AO,5) • delete(AO) • clear AO • %% clean up, close down • %% Now let's plot the function for 5 cycles • x = 0:.1:2*pi*5; • data = tan(sin(x)) - sin(tan(x)); • plot(x,data) • %% Now let's add random noise • %y = awgn(data,10,'measured'); % Add white Gaussian noise. • y = sin(x)'; • plot(x,data,x,y) % Plot both signals. • legend('Original signal','Signal with AWGN');