Suppose we have the following discrete time signal: x[n] = 2(0.8)¹u[n - 1] Let X10 [k] represent the 10-point DFT of x[n] over the range n=0,1,2,...,9 and define x 10 [n] as the inverse DFT of X10 [k]. (a) Find X10 [0] (b) Find 10 [5], 10[10], 10[15], 10[20], 10 [25] Now suppose we have the following discrete time signal: w[n] = (2(0.8)-u[n-1] 0 ≤ n < 10 0 otherwise Let W20[k] represent the 20-point DFT of w[n] and define W20 [n] as the inverse DFT of W20 [k]. (c) Find W20 [0]

6. Consider the circuit depieted in the figure below, R V,eiut where we shall assume that the input voltage can be written as Vin(t) = Re{Voet} with complex-valued Vg, and likewise that the output voltage Vout(t) = Re{Vje*} with complex-valued Vị. With j = V-1, the complex impedances of resistors, ca- pacitors, and inductors in this case are respectively given by ZR = R, Zc = ZL = jwL. (a) Determine Vı/Vo as a function the angular frequency w given the parameters R and C. (b) Evaluate Vi/Vo in the limit w +0 and in the limit w + o0. Is the circuit above a low-pass filter, a high-pass filter, or something else? w

a) Using a NTC thermistor with "B=4000K, resistance: 5k2 at 25°C" we want to measure the temperature of an environment in the range 50°C - 100°C. Design a suitable circuit for this purpose with "zero" output signal |v) value at 25ºC. Express the output and its temperature-sensitivity in terms of temperature (y=f(T°C) and dy/dT= f(TºC)). Roughly draw y and dy/dT for the given temperature range. (Note: For NTC: RNTC=Roexp(B(1/T-1/To))