Question

A simplified transfer function governing a system is given by function ℎ(?), which we will represent as a single decaying exponential with ?= 3?? and time delay of 1 ??. Consider a burst release of neurotransmitters represented by the periodic delta function input ?(?) occurring at a rate of 250?? over 20??.

a) Calculate the system response ?(?) by hand.
b) In MATLAB, plot the system response, ?(?).

Answer 1

-0.001 h(t) = e 0.003

Time period T of x(t) = 1/(250 Hz) = 0.004 s

Therefore, no. of impulses between 0 ans 20 ms = 0.020/0.004 + 1 =6 [ +1 to include the impulse at 0 ms]

time of occurrence of the impulses = {0, 0.004, 0.008, 0.012, 0.016, 0.020}

h(t) *r(t)

= {h(t) * 5(t), h(t)(t - 0.004), ....,h(t) * 5 t – 0.020)}

[from the convolution property of unit impulse function]

= {h(0), (0.004), ...., h(0.020)}

= {e _0-0.001 0.003 e 0.004-0.001 0.003 e 0.008-0.001 0.003 e 0.012-0.001 0.003 e 0.016-0.001 0.003 e 0.020-0.001 0.003

= {1.3956,0.3679, 0.0970,0.0256, 0.0067, 0.0018}

MATLAB code:

t=0:0.004:0.020;

y=exp(-(t-0.001)/0.003);

figure;stem(t,y)

Edit View Eile Da Insert m Figure 2 (on gpuwks) Tools Desktop Window Help 28. E - O 1.5 y(t) = h(t)*x(t) ---> 0 0.004 0.02 0.008 0.0120.016 t--->