1. Introduction to Signals Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

2. Introduction to Signals - Hany Ferdinando2 General Overview This gives students an introduction about signal and the notation used in this course It discusses time signals with elementary operation on and among signals Students learn the generalized signals

3. Introduction to Signals - Hany Ferdinando3 Definition Signal is a phenomenon that represents information Since any signals always is one of a collection of several of many possible signals, signals may mathematically be represented as elements of a set, called signal set

4. Introduction to Signals - Hany Ferdinando4 Example… Human vocal mechanism produces speech by creating fluctuation in acoustic pressure Monochromatic picture consists of variation patterns in brightness

5. Introduction to Signals - Hany Ferdinando5 Signals The signals we are interested in are functions of a variable (usually time) Domain of a signal is a subset T of the real line and is called signal axis The signal may take values in any set A, called the signal range

6. Introduction to Signals - Hany Ferdinando6 Signals x: T  A means a signal with signal axis T and signal range A It can be written also as x: A T if T is in time, then the signal is called time signal, and the signal axis is called time axis Another signal is frequency signal

7. Introduction to Signals - Hany Ferdinando7 Various kinds of signals Discrete- and continuous-time signals Finite- and infinite-time signals Periodic and Harmonic signals

8. Introduction to Signals - Hany Ferdinando8 Discrete-time Signal Time axis T is discrete if it consists of a finite or countable set of time instant Discrete-time signals is written as x(n), n is integer Example:  sampled signal from ADC  The number of product in one hour production

9. Introduction to Signals - Hany Ferdinando9 Continuous-time Signal Time axis T is continuous is if it consists of an interval of R (real number) Continuous-time signal is written as x(t), t is real number Example:  x(t) is electric signal in a circuit  Temperature measurement in a room

10. Introduction to Signals - Hany Ferdinando10 Finite-, Semi-infinite-, Infinite-time Finite-time axis: if a time axis is contained in a finite interval, it is called a finite axis Semi-infinite-time axis: if a time axis is bounded from the left, it is called right semi-infinite time axis, and vice versa Infinite-time axis: it is neither bounded from the left nor from the right

11. Introduction to Signals - Hany Ferdinando11 The Unit Impulse  (n) = 1, for n = 0 0, otherwise T is discrete-time axis 0 1 -22

12. Introduction to Signals - Hany Ferdinando12 The Unit Step Signal u(n) = 1, for n ≥ 0 0, for n < 0 u(t) = 1, for t > 0 0, for t < 0 … n=0 t =0

13. Introduction to Signals - Hany Ferdinando13 Remark!! and This is only for discrete-time signal!!

14. Introduction to Signals - Hany Ferdinando14 The Unit Ramp Signal ramp(n) = n, for n ≥ 0 0, for n < 0 ramp(t) = t, for t ≥ 0 0, for t < 0

15. Introduction to Signals - Hany Ferdinando15 Rectangular & Triangular Pulse rect(t) = 1, for -0.5 ≤ t ≤ 0.5 0, otherwise trian(t) = 1-|t|, for |t| < 1 0, otherwise

16. Introduction to Signals - Hany Ferdinando16 Complex Exponential Signal atat a > 10 < a < 1

17. Introduction to Signals - Hany Ferdinando17 Complex Exponential Signal a > 10 < a < 1 anan

18. Introduction to Signals - Hany Ferdinando18 Complex Exponential Signal a < -1-1 < a < 0 anan

19. Introduction to Signals - Hany Ferdinando19 Periodic Signals Definition: a signal that repeats itself indefinitely The length of time after which the signal starts repeating itself is called period x(t + T) = x(t) for all T is the period

20. Introduction to Signals - Hany Ferdinando20 Harmonic Signals x(t) = e j  t or x(n) = e j  n  and  = 2  f is called angular frequency of the harmonic The signal of the form x(t) = a e j  t a is called the complex amplitude of the harmonic signal If T is the period, then e j  T = 1

21. Introduction to Signals - Hany Ferdinando21 Periodicity of harmonic signal Periodicity of continuous-time harmonic signal is simple. it is formulated as T = 1/|f| Periodicity of discrete-time harmonic signal is a little more complicated. The reason is sampling two continuous-time signal with different freq. may result in the same discrete-time signal. This phenomenon is called aliasing. See next figure…

22. Introduction to Signals - Hany Ferdinando22 Periodicity of harmonic signal

23. Introduction to Signals - Hany Ferdinando23 Periodicity of harmonic signal Discrete-time signal with  o varies from 0 to 2  will show this ‘strange’ situation. Try it using Matlab!! 1

24. Introduction to Signals - Hany Ferdinando24 Periodicity of harmonic signal In order for the signal complex exponential to be periodic with period N > 0,  o N must be multiple of 2  Period

25. Introduction to Signals - Hany Ferdinando25 Compare…

26. Introduction to Signals - Hany Ferdinando26 Elementary Operations Signal range transformation  Amplification or attenuation  Quantization Time transformation  Time expansion, time compression and time reversal  Time translation

27. Introduction to Signals - Hany Ferdinando27 Amplification and Attenuation If |  | > 1 then it is an amplification If |  | < 1 then it is an attenuation y(t) =  x(t) The time axis remains the same while the range is changed!

28. Introduction to Signals - Hany Ferdinando28 Quantization This is an important range transformation This transformation is needed when signals are processed by a digital computer or other digital equipment The Analog to Digital Converter (ADC) does this operation

29. Introduction to Signals - Hany Ferdinando29 Expansion, compression and reversal If 0 <  < 1 then it is time expansion If  > 1 then it is time compression if  = -1 then it is time reversal or reflection y(t) = x(  t)

30. Introduction to Signals - Hany Ferdinando30 Translation y(t) = x (t –  ) shift x(t) to right (delay) by  y(t) = x (t +  ) shift x(t) to left by  What about y(t) = x(-t –  ) and y(t) = x(-t +  ) ???

31. Introduction to Signals - Hany Ferdinando31 Translation and Reflection We can get x(t-  ) from x(-(-t+  )) We can get x(-t-  ) from x(-(t+  )) We can get x(t+  ) from x(-(-t-  )) We can get x(-t+  ) from x(-(t-  )) Check these statements!!

32. Introduction to Signals - Hany Ferdinando32 Next… Signals and System by Alan V. Oppenheim, chapter 2, p 35-45 Signals and Linear Systems by Robert A. Gabel, chapter 2, p 23-37, chapter 3, p 121-127 An introduction to signals is discussed! Students should do some assignment either from the lecturer or from the books. For the next meeting, please prepare yourself by reading chapter about system.