Generating smoothed versions of square wave, triangular, etc

Question

I'm trying to get functions depicting smoothed out versions of wave forms(triangle,square, sawtooth, reverse sawtooth) which have the same amplitude and frequency as $cos(x)$.

I found a few trigonometric functions depicting such wave forms:

$\delta=0.05$,

$Square_{smooth}=2.1\tan^{-1}(\sin(\pi x/3)/\delta)/\pi$

$Triangle_{smooth}=1.25-5\cos^{-1}((1-\delta)\sin(\pi x/3.2))/(2\pi)$

$h=12.5$,

$ Sawtooth_{smooth}=-1.5 + 3((x/6-0.5) -(\tanh((((x/6-0.5)+0.5)-\lfloor{(x/6-0.5)+0.5}\rfloor-0.5)h)/(2\tanh(0.5h))+\lfloor{(x/6-0.5)+0.5)}\rfloor-0.5))$

but they do not have the same amplitude or frequency unfortunately. I think $-cos(x)$ might work for $Triangle_{smooth}$, but I'm still looking for ways to fix $Square_{smooth}$ and $Sawtooth_{smooth}$, anyone have any ideas?

Answer 1

So, a "smooth" (differentiable, without Gibbs effects) square wave can be defined as $$s_{ss}(t)=\frac{2A}{\pi}\arctan\left[\frac{\sin(2\pi tf)}{\delta}\right].$$ (Source.) The parameter $A$ controls the amplitude, and $f$ the frequency. For example, using Matlab with $A=1.5$ and $f=2$:

t = 0:1/100:2;
delta = 0.01;
A = 1;
f = 2;
smoothsq = (2*A/pi)*atan(sin(2*pi*t*f)/delta);
plot(t,smoothsq);
axis([-0.2 2.2 -1.6 1.6]);

produces the plot:

Something similar should work for the other waveforms.

Answer 2

It seems the amplitude is not scaled properly. Rather than (2*A/pi) using (A/atan(1/delta)) seems more appropriate. In other words I propose:

y = (A/atan(1/delta))*atan(sin(2*pi*t*f)/delta);

Below is a figure illustrating the difference between the two scaling approaches. For low delta values the difference is not clear but for high delta values the amplitude departs from what is desired. The images can be reproduced using the MATLAB code below.

t=linspace(0,2*pi,500);
delta = 0.1;
A = 1;
f = 1/(2*pi);
y1 = A*sin(2*pi*t*f);
y2 = (2*A/pi)*atan(sin(2*pi*t*f)/delta);
y3 = (A/atan(1/delta))*atan(sin(2*pi*t*f)/delta);

figure; hold on;
h1=plot(t,y1,'r-','LineWidth',5);
h2=plot(t,y2,'g-','LineWidth',4);
h3=plot(t,y3,'b-','LineWidth',2);
legend([h1 h2 h3],{'Sine','Smooth square wave','Corrected smooth square wave'},'Location','SouthOutSide');
axis tight; axis square;
set(gca,'FontSize',15); grid on; box on;
drawnow;