Unit 1 -Introduction to signals and standard signals

Unit 1 -Introduction to signals and standard signals

1. Introduction of Signals & S t andard Signals Dr.K.G.SHANTHI Professor/ECE [email protected] 1 RMK College of Engineering and Technology

2. Introduction of Signals Types of Signals Basic Standard signals Applications 2 Contents

3. Introduction of signals

4. Signal A function of one or more variables that convey information on the nature of a physical phenomenon. Signals are represented mathematically as functions of one or more independent variables. The independent variable: time (speech),space (images). Eg . A speech signal can be represented mathematically by acoustic pressure as a function of time 4

5. Signals Signal is denoted by x(t), where the variable t is called independent variable and the value x(t) is dependent variable. Dependent variable = voltage Independent variable = time

6. Signal Classification :Based on Dimension 1 Dimensional - Varies with respect to one independent variable ( time or space or distance ) e.g. Speech, daily maximum temperature, annual rainfall at a place 2 Dimensional - Varies with respect to more than one independent variable e.g . Image 3 Dimensional - Varies with respect to more than two independent variables ( space ,distance and depth ) e.g. Video I Dimensional 3 Dimensional 2 Dimensional

7. Basic types of signals Continuous Time Signal (CT) Discrete Time Signal (DT) (CT) (DT)

8. Continuous Time Signal (CT) The independent variable is continuous, and thus these signals are defined for a continuum of values of the independent variable. A signal that is defined for every instants of time is known as continuous time signal. Continuous time signals are continuous in amplitude and continuous in time. A speech signal as a function of time and atmospheric pressure as a function of altitude are examples of continuous-time signals. It is denoted by x(t) 8 Eg : Human Voice, TV signal, Analog Phone, Record player

9. Discrete Time Signal (DT) 9 A signal that is defined for discrete instants of time is known as discrete time signal. Eg:stock market index Discrete time signals are continuous in amplitude and discrete in time. It is also obtained by sampling a continuous time signal. It is denoted by x[n] Eg : Computers, Digital Phone, VCD,DVD,2G/3G cellular phone

10. Sampling Discrete-time signals are often obtained by sampling continuous-time signals 10 Where, T = Time between samples

11. Basic standard signals

12. Basic (Elementary or Standard) signals Step Signal Ramp signal Impulse signal Parabolic Signal Sinusoidal and exponential signal Sinc signal Rectangular signal Signum signal Triangular signal 12 CT and DT Test Signals Input Signals for process

13. 13 Unit Step Signal STEP SIGNAL Represented by u(t ) and u(n) Heaviside  function Application : DC Generator(Switching on and off of a device) Communication applications x(t)=A for t≥0 x(t)=0 for t<0 x(n)=A for n≥0 x(n)=0 for n<0 DT Signal CT Signal

14. 14 RAMP SIGNAL Represented by r(t) and r(n) Application: Current and Voltage relation circuits DT Signal CT Signal

15. 15 parabolic SIGNAL Acceleration function Represented as p(t) and p(n) Application: The Bike responds to acceleration x(t)=At 2 /2;t≥0 x(t)=0 for t<0 DT Signal CT Signal

16. Relation between Unit Step signal, Unit ramp signal and Unit Parabolic signal: Unit ramp signal is obtained by integrating unit step signal ∫u(t) dt =∫1dt=t=r(t) Unit Parabolic signal is obtained by integrating unit ramp signal ∫r(t) dt =∫ tdt =t 2 /2=p(t) Unit step signal is obtained by differentiating unit ramp signal d/ dt r(t)=d/ dt (t)=1=u(t) Unit ramp signal is obtained by differentiating unit Parabolic signal d/ dt p(t)=d/ dt (t 2 /2)=t=r(t) 16

17. Pulse signal (Rectangular pulse function) Pi Function, Gate function Represented by rect (t) 17 x(t)=A;t 1 ≤t≤t 2 and x(t)=0 elsewhere Unit Pulse signal π(t)=1;|t|≤1/2 and π(t)=0;elsewhere x(n)=A;n 1 ≤n≤n 2 and x(n)=0 elsewhere DT Signal CT Signal

18. 18 IMPULSE SIGNAL Delta  function CT - Dirac delta function DT- Kronecker delta function Unit Area signal Application Thunderbolt ECG function DT Signal CT Signal

19. 19 SINUSOIDAL SIGNAL Cosinusoidal signal-CT Ω=2πf=2π/T and Ω is angular frequency in rad /sec f is frequency in cycles/sec or Hertz A is amplitude T is time period in seconds 𝛷 is phase angle in radians Application Any sound signal The light signal Sinusoidal signal-CT Sinusoidal signal -CT Cosinusoidal signal -DT

20. 20 DT Signal Signum SIGNAL Represented by sgn (t) Used in Communication CT Signal

21. 21 DT Signal Triangular SIGNAL Represented by tri(t) Application Analog to Digital conversion circuits CT Signal

22. 22 DT Signal Sinc SIGNAL Sine Cardinal function Represented by sinc (t) Used in Digital Signal Processing Information Theory , t = 0 1 , t = 0 CT Signal

23. 23 Exponential SIGNAL Applying Euler’s Identity 𝑥(𝑡) =𝐴𝑒 𝑠𝑡 =𝐴𝑒 (𝜎+𝑗 Ω 𝑡) =𝐴𝑒 𝜎𝑡 𝑒 𝑗 Ω 𝑡 Complex exponential signal is defined as 𝑥(𝑡) =𝐴𝑒 𝑠𝑡 where 𝐴 is amplitude, s is complex variable 𝑠=𝜎+𝑗 Ω 𝑥(𝑡) =𝐴𝑒 𝜎𝑡 (𝑐𝑜𝑠 Ω 𝑡+𝑗𝑠𝑖𝑛 Ω 𝑡)

24. 24 Complex Exponential SIGNAL Where Then 𝑥 (𝑡)=𝐴𝑒 𝜎𝑡 (𝑐𝑜𝑠 Ω 𝑡+𝑗𝑠𝑖𝑛 Ω 𝑡), When , 𝜎 = + ve 𝑥 𝑟 (𝑡) =𝐴𝑒 𝜎𝑡 𝑐𝑜𝑠 Ω 𝑡 𝑎𝑛𝑑 𝑥 i (𝑡) =𝐴𝑒 𝜎𝑡 𝑠𝑖𝑛 Ω 𝑡 Exponentially growing sinusoidal signal Exponentially growing cosinusoidal signal Exponentially decaying cosinusoidal signal Exponentially decaying sinusoidal signal Where Then 𝑥 (𝑡)=𝐴𝑒 -𝜎𝑡 (𝑐𝑜𝑠 Ω 𝑡+𝑗𝑠𝑖𝑛 Ω 𝑡), When , 𝜎 = - ve 𝑥 𝑟 (𝑡) = 𝐴𝑒 -𝜎𝑡 𝑐𝑜𝑠 Ω 𝑡 𝑎𝑛𝑑 𝑥 i (𝑡) =𝐴𝑒 -𝜎𝑡 𝑠𝑖𝑛 Ω 𝑡

25. Real Exponential SIGNAL 25 Real Exponential signal is defined as 𝑥(𝑡) =𝐴𝑒 𝜎𝑡 where A is amplitude. It is obtained when Ω =0. Depending on the value of ‘𝜎’ we get dc signal or growing exponential signal or decaying exponential signal

26. 26 Problem Consider 𝜎 =1 (𝜎 >0 ) x(t) = e 𝜎𝑡= e 𝑡 t= 0, e = 1 t= 1, e 1 = 2.7 t= 2, e 2 = 7.3 Consider 𝜎 =- 1 (𝜎 < 0) x(t) = e 𝜎t= e -t t= 0, e = 1 t= 1, e -1 = 0.36 t= 2, e -2 = 0.13 Rising Falling

27. Real Exponential signal is defined as 27 Exponential SIGNAL- DT x(n) = a n for all n 0<a <1 = Sequence decays exponentially a >1 = Sequence grows exponentially a <0 = Discrete time exponential signal takes alternating signs Decreasing exponential signal Increasing exponential signal Increasing exponential signal with alternating signs Decreasing exponential signal with alternating signs

28. 28 Exponential SIGNAL Consider a = ½ (0< a <1 ) n= 0, (½) = 1 n= 1, (½) 1 = ½ = 0.5 n= 2, (½) 2 = ¼ = 0.25 Consider a = 2 ( a >1) n= 0, 2 = 1 n= 1, 2 1 = 2 n= 2, 2 2 = 4

29. 29 Practise a = -2 a = -½ Consider a = -½ n= -2, (-½) -2 = 4 n= -1, (-½) -1 = -2 n= 0, (-½) = 1 n= 1, (-½) 1 = -½ = -0.5 n= 2, (-½) 2 = ¼ = 0.25 Consider a = - 2 n= -2, (-2) -2 = 1/4 n= -1, (-2) -1 = -1/2 n= 0, (-2) = 1 n= 1, (-2) 1 = -2 n= 2, (-2) 2 = 4 n= 3, (-2) 3 = -8 -2 -1 0 1 2 3 x(n) = (-2) n 1/4 -1/2 1 -2 4 -8 n w.k.t X(n) = α n -2 -1 1 2 x(n)= (-1/2) n 1/4 -1/2 1 -2 4 n

30. Complex Exponential SIGNAL- Dt 30 Complex Exponential signal is defined as Where Exponentially decreasing Cosinusoidal signal Exponentially decreasing sinusoidal signal Exponentially growing Cosinusoidal signal Exponentially growing sinusoidal signal

31. 31 Thanks!