1.55. Evaluate the following integrals: (a) f(cos t)u(t) dt (b) S(cos t)8(t) dt (c) (cos t)u(t – 1)8(t) dt (d) t sin-8(1 – t) dt |
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- The mathematical models of some continuous time systems are given above. x(t) is the input signal of the system, and y(t) is the output signal of the system. Which of the following systems is linear and which is nonlinear?It is given that v(t)=20 sin π|t|V over the interval –1≤t≤1 s. Thefunction then repeats itself. Does the function have half-wave symmetry?It is given that f(t)=4t2 over the interval –2<t<2 s. Is the function even or odd?
- a) Y(ω) = 1/(5+jω) The expression above defines the frequency domain signal Y(ω) from a time domain signal y(t) i) Determine the imaginary & real parts of Y(ω). ii.Determine the signal y(t) that produced the signal Y(ω). b)What are the 3 areas of application of i.Signals ii.Systems c) Briefly summarise with explanation the difference between discrete & continuous time signal and also write an expression that describes each signal type.1. Find the area under the function shown.. 2. What is the duration of the function when ϵ=0? 3. What is the magnitude of f(0) when ϵ=0?a) Show whether the following signals are periodic, if so find the fundamental periods. x [n] = 4 * cos (0.3π * n) + 3 * sin ^ 2 (0.2π * n) x (t) = 2 * cos (10π ^ 2 * t) + sin (3π * t) b) Show whether the system given the input-output relation y (t) = βx (t) + u (t) is linear, memoryless, time invariant, causal and stable. Here β is constant.
- Step functions can be used to define a window function. Thusu(t −1)−u(t−4) defines a window 1 unit high and 3 units wide located on the time axis between 1 and 4. A function f(t) is defined as follows: f(t)=0,t≤0;=−20t,0≤t≤1 s= −20,1 s≤t≤2 s=2 cosπ2t,2 s≤t≤4 s;=100−20t,4 s≤t≤5 s=0,5 s≤t<∞. 1. Sketch f(t) over the interval −1 s≤t≤6 s. 2. Use the concept of the window function to write an expression for f(t).When the unit step function is applied to the input of the system whose block diagram is given below, the output response takes the value c (0.2) = 0.11 for t = 0.2 s and c (infinity) = 0.333 for t = infinity. What is the steady-state error of the system? calculate.Find the steady-state difference between the input f (t) and the response x(t), iff (t) = 6t. ẍ + 8ẋ + x = f (t) x(0) = ẋ (0) = 0
- Consider a system whose input x (t) and output y (t) satisfy the differential equation of the first order dy(t)/dt + 2y(t) = x(t) The system also satisfies the initial rest condition. Determine the output of the system y (t) when the input is x (t) = e ^3t u(t)Suppose an electrical signal generated, y(t) can be modelled as follows where t, is time in second, y(t) is voltage in mv. a) Produce graphs showing the amplitude and phase spectrum of y(t). b) State whether this time-domain signal y(t) is periodical, or non-periodical and justify your answer.A)Perform the N=5 point circular convolution of x and h using DFT. B)Perform the N=5 point circular convolution of x and h in time-domain.