# Square Wave transfer function

How could i determine a transfer function (A(s)) which responds to a symmetrical 1Vp-p 1kHz square-wave input signal, and gives a 5Vp-p square-wave output signal

with tilt==> δ=10% and rise/fall time ==>tr=tf=10μs

• So you want to ADD a 10% tilt (which needs a HighPassFilter) and has the ability to slow a very fast input edge down to 1-pole response time of 10uSeconds? – analogsystemsrf May 30 '19 at 5:08
• 10% tilt across 1milliSecond, implies a timeconstant of about 7 milliSeconds, which (1/(7milliSecond * 6.28) = 1/45 milliirad = 22 Hertz HPF. – analogsystemsrf May 30 '19 at 5:10
• @analogsystemsrf exactly for the high pass but the rise and fall time can be a 1 pole low pass filter right?, my current transfer function is a A(s) = 5*(s/s+200)*(220k/s+220k) getting a 5 gain and a high freq pole due to rise and fall at 220k rad/s , low freq pole at 200rad/s due to tilt, but i was just wondering if it is correct – forthelulx May 30 '19 at 5:13
• I see we got the same answer – Tony Stewart Sunnyskyguy EE75 May 31 '19 at 3:56

A 1st order approximation using :

For a 1kHz square wave with T/2 sag or tilt by 10%
$$\\tau_1=5ms =0.5ms/ 0.1(sag)= 0.5ms/0.1 = 1/\omega_1\$$ = HPF

Sine tr=0.35/f_{-3dB} BW for rise time 10 to 90%

$$\\tau_2=4.55us=1/\omega_2 =t_r/(2\pi *0.35)=10us/(2\pi *0.35)\$$ = LPF assuming 10~90% for rise time

Then the transfer function can be made with gain constant =5

$$\H(s)=5\dfrac{0.005s+1}{(4.55*10^{-6}) s+1}\$$

Take the Laplace transform of the input and output signals, and divide one by the other. In this particular case the signals are piecewise linear.

For (simple!) example, if the input is a unit step, and the desired output is a unit ramp for 1 sec, followed by a constant value of unity, the input and output signals would be: $$\\frac{1}{s}\$$ and $$\\frac{1-e^{-s}}{s^2}\$$, respectively, and the required TF would be: $$\\frac{1-e^{-s}}{s}\$$.

In some cases the resultant TF may not be physically realisable, since term(s) in $$\\small e^{-sT}\$$ may arise in the TF denominator. I suspect this is the case for your application. Approximations, such as a Pade approximant, may then prove useful.