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, , 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 The 2 nd order diff eqn can be solved using characteristic equation or auxiliary equation Remember: v 1 -y = i R2  v 1 =i R2 +y and i(t) =C dv/dt Find the mathematical relationship in terms of input & output Substitute:

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

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');