Transcription of EC6303 Signals and Systems Department of ECE 2016-2017 ...
1 EC6303 Signals and Systems Department of ECE 2016-2017 UNIT I -CLASSIFICATION OF Signals AND Systems PART A 1. What are the major classifications of Signals ? Signals are classified as Continuous Time (CT) and discrete Time(DT) Signals . Both CT and DT Signals are further classified as Deterministic and Random Signals , Even and Odd Signals , Energy and Power Signals , Periodic and Aperiodic Signals 2. With suitable examples distinguish a deterministic signal from a random signal . Define a random signal .(May 2013) Deterministic signal : A signal which can be modeled (represented) by a mathematical equation. Example: cosine signal Random signal : A signal which cannot be modeled by a mathematical equation is called random signal . Example: Speech signal 3. Define energy signal and power signal .(April 2015) A signal x(t) is said to be energy signal if , Energy is finite 0 < E < and average power is zero P=0 Where E = energy and P = Average power < and energy is infinite A signal x(t) is said to be power signal if power is finite 0 < P E = where E = energy and P = Average power 4.
2 Give the mathematical and graphical representation of unit step sequence. 5. Give the mathematical and graphical representation of unit ramp signal . 6. What are periodic Signals ? Give example. A signal x (t ) is said to be periodic if x(t) x(t T ) for all t . The smallest value of T for which the condition is satisfied is called the fundamental : sinusoidal Signals 7. What is odd signal ? Give example. A signal x ( t ) is said to be odd signal if x (t ) = x( t) x ( t ) .Example: x (t ) Asin t 8. State two properties of unit impulse function. (Dec2014) 1. (t) is the limit of graphs of area 1, the area under its graph is 1. (t)peak response at origin. 9. Define symmetric and anti symmetric signal . Symmetric signal :It is a even signal ,A signal x (t ) is said to be symmetric signal if x ( t ) = x (t ) . Example: x (t ) Acos t Anti symmetric signal : Dhanalakshmi Srinivasan College of Engineering and Technology 1 EC6303 Signals and Systems Department of ECE 2016-2017 A signal x (t ) is said to be anti symmetric signal if x (t ) = x ( t ).
3 Example: x (t ) Asin t 10. Verify whether x(t) Ae atu(t) , a 0 is an energy signal or not. x(t) Ae atu(t) , a 0 T 2 T 2 e 2a t T A 2 : Energy lt x(t) dt lt Ae a t dt lt A2 Joules T T T 2a 0 2a T 0 1 T 2 1 e 2at T 0 Watt power lt x(t) dt lt A2 T 2T T 2T 2a T 0 Energy is finite, Power is zero. The signal is energy signal 11. Show that the complex exponential signal is periodic and that the fundamental period is x(t ) e j 0 t : x(t T ) e j 0 (t T ) e j 0 t e j 0T x(t ) x(t T ) when e j 0T 1 when 0T 2 m where m int eger smallest value for m 1, fundamental period = 2 / 0 12.
4 Determine the power and RMS value of the signal x(t) e jat cos t . o 1 T 2 1 T 1 cos 2 0 t dt power lt e j a t cos 0 t dt lt sin ce e jat 1 T 2T T 4T T T lt 1 (2T ) 1 watt ; RMS value 1 2 2 T 4T the average power of the signal u(n) u(n N ). Average power of a DT signal x(n) is 1 N 2 1 N Power lt x(n) lt (1)2 0 watt N (2N 1) n N N (2N 1) n 0 14. Draw the following signal x ( t ) = [ u ( t ) u ( t 10) ] .(Nov2014) 15. Give the formula for decomposing an arbitrary signal x(t) in to odd and even part.
5 CT signal : x(t) xe (t) x o (t) : x e (t) even component & x o (t) odd component x e (t) 1 x(t) x ( t) ; xo (t) 1 x(t) x ( t ) 2 2 DT signal : x(n) xe (n) x o (n) : x e (n) even component & x o (n) odd component x e (n) 1 x(n) x ( n) ; xo (n) 1 x(n) x ( n ) 2 2 16. Give x[n] = {1,-4,3,1,5,2}. Represent x[n] interms of weighted shifted impulse functions.(May2014) x[n] can be represented interms of its weighted shifted impulse functions as follows: x[n]= [n] - 4 [n-1]+3 [n-2]+ [n-3]+5 [n-4]+2 [n-5]. 17. Distinguish static system from dynamic system. Static system: Static system is a system with no memory or energy storage element. Output of a static system at any specific time depends on the input at that particular time. Dynamic system: Dynamic Systems have memory or energy storage elements. Output of a dynamic system at any specific time depends on the inputs at that specific time and at other times.
6 18. Define a time invariant system. A system is said to be time invariant if its input-output characteristics do not change with time. Dhanalakshmi Srinivasan College of Engineering and Technology 2 EC6303 Signals and Systems Department of ECE 2016-2017 Let y(t) x(t) ; denotes some transformation (operation) on x(t) ;x(t)-input,y(t)- output Let y (t, T ) denote the output due to delayed input x (t T ) , y(t , T ) x(t T ) let y( t T ) be the out put delayed by T if y(t T ) y( t , T ) then the system is time invariant 19. Define a continuous time LTI Give the conditions for a system to be LTI system. (Dec2013) A continuous time system which posses two properties i) linearity (Obeys superposition principle) ii) Time invariance(Input output characteristics do not vary with time) is a CT LTI system. 20. Determine whether the system described by the following input-output relationship is linear and causal y(t) = x(-t) y(t) x ( t ) input output relationship y(t) output & x(t) input Checking for linearity: For an input x1 (t) , the output y1 (t) is, y1 (t) x1 ( t) For an input x2 (t) , the output y2 (t) is, y2 (t) x2 ( t) For an input a x1 (t) b x2 (t) , the output y3 (t) is, y3 (t) a x1 ( t) b x2 ( t) y3 (t) a y1 (t) b y2 (t) The system obeys superposition principle.
7 Therefore the system is linear Checking for causality: For t 1, y ( 1) x (1 ) For negative values of time t , the output depends on the future input. Therefore the system is non-causal. 21. Determine whether the following signal is energy or power signal . And calculate its energy or power x(t)=e-2tu(t).(Dec 2012) T 2 T 2 e 4 t T 1 Energy T lt x(t) dt T lt e 2 t dt lt Joules T 4 T 0 4 0 1 T 2 1 e 4t T power lt T x(t) dt lt 0 Watt T 2T T 2T 4 0 Energy is finite,Power is zero. The signal is energy signal 22. Check whether the following system is static (or) dynamic and also causal (or) non- causal:y(n)=x(2n) (Dec 2012) For a given n the output depends on the future input. Therefore the system is non-causal. The system is a dynamic system. 23. Sketch the following signal x(t)=2t and x(n)=x(2n-3)(May 2014) 6 n the fundamental period of the given signal x(n) sin 1 (May2012) 7 6 ; N 2 k ; N k 2 7 k 14 where k , N are int eger 0 0 6 7 6 for some value of k, N takes the lowest possible int eger value when k 6, N 14 x(n) is periodic with period N=14 25.
8 Verify whether the system described by the equation is linear and time (t)=x(t2) Linearity: y t x(t 2 ) ; y(t) T[x(t)] x(t 2 ) For an input x1(t), y1 (t) T[x1 (t)] x1 (t 2 ) Dhanalakshmi Srinivasan College of Engineering and Technology 3 EC6303 Signals and Systems Department of ECE 2016-2017 For an input x2(t), y2 (t) T[x2 (t)] x2 (t 2 ) Weighted sum of outputs is given by ay (t) by 2 (t) ax (t 2 ) bx 2 (t 2 ) 1 1 Output due to weighted sum of inputs is y 3 (t) T[ax (t) bx 2 (t)] [ax (t 2 ) bx 2 (t 2 )] 1 1 Therefore, the system is linear. Time invariance: y(t) x(t 2 ) , y(t) T[x(t)] x(t 2 ) If the input is delayed by k units of time then the output is , y(t, k) T[x(t k)] x((t k)2 ) Output delayed by k units of time is, y(t k) x(t 2 k), y(t, k) y(t k) Therefore, the system is time variant. 26. Give the relationship between unit impulse function (t) , step function u(t) and ramp function r(t).
9 (Nov 2015) (t ) d u(t ) ,u(t ) d r (t ) dt dt 27. How the impulse response of a discrete time system is useful in determining stability and causality?(April 2015) For a causal system impulse response h(n)=0 for n 0 For stable system system function poles located inside unit circle. PART B 1. i) Sketch the signal x (t) = u (t) u ( t - 15). Determine the energy and power in the signal x(t). 1 n ii) Determine the energy and power in the signal )n(u )n(x 2. i) How are Signals classified. ii) Determine whether the following signal is periodic. If periodic, determine the fundamental period: x t 3cos(t) 4 cos(t / 3) (Dec 2012) iii) Give the equation and draw the waveforms of discrete time real and complex exponential Signals . 3. i) Define an energy and power signal (4) ii) Determine whether the following Signals are energy or power and calculate their energy or power.(12). (May 2013) 1 n t 2 (i) x(n) = u(n) ( ii) x(t) = rect (iii)x(t) = cos ( ot) 2 To 4.
10 I) Define unit step, Ramp, Pulse, Impulse and exponential Signals . Obtain the relationship between the unit step function and unit ramp function. (10) n n n ii) Find fundamental period x(n) cos sin cos (May 2013) 2 8 4 3 5. Determine whether the discrete time system y(n)=x(n)cos( n) is (i) memoryless (ii) Stable (iii) causal (iv) linear (v)time invariant.(Dec 2013) 6. i) Determine whether the signal x(t)=sin20 t+sin5 t is periodic and if it is periodic find the fundamental period(5) (Dec 2013) ii) Discuss various forms of real & complex exponential Signals with graphical representations (6) (Dec 2013) iii) State the precedence rule for combined time scaling and time shifting (5) 7. Check whether the system is linear, causal, time invariant and or stable (Dec2014) i) y(n) = x(n) - x(n-1) ii) y(t) d x(t) dt 8. Check whether the following Signals are periodic/aperiodic Signals .(Dec2014) t n x(t) cos 2t sin x(n) 3 cos cos 2n 5 2 9.
