2 problems : TTL and logic inversion
I see you are using two AND gates when he indicated one AND for positive logic output = ON low side Nch then OR for negative logic high side Pch.
But if you are using dual Nch half Bridge with PWM low side to Boost Vs on high side then you may want positive logic for both.
Then you need to invert the top side to a NOR with AND on the low side.
Using de Morgan's Law there are many other variations. The only problem with Andy's XOR gate is the prop delays at high speeds (> 1MHz) are unequal for tHL and tLH.
Falstad Sim
Due to source current TTL does not work for this method using high resistor values for active low.
It is designed for CMOS logic which is prone to +/-33% tolerance on the threshold between devices and all temperatures and thus must be taken into consideration. CMOS source impedance is defined by Vol/Iol {typ, max} and change with each family from HC to AC but range 66 to 22 Ohms unless CD4xxx HV logic family was used. Schmitt triggers are prone to the same threshold tolerances. Thus high values of R and small values of C are practical as input impedance > 10M and < 5pF roughly.

Middle plot show both combined to for easy display of overlap.
The concept is slow turn ON and fast turn OFF to create the deadtime signals.