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What do you want to know exactly?
The meaning of the lines listed in the beginning or the meaning of the whole demo?
I can't understand your question, since the operation involved with the listed lines is quite simple, even if you had no knowledge of Verilog syntax.
On the other hand I think the purpose of the whole code is described by the title of you post: it generates a sort of 'breathing' behaviour in the LEDs, by dimming the brightness through pulse width modulation.
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idon't understand about value in if condition ,PWM_Adj
1."
if(counter[26]) " when condition this active? i don't know value in counter to reach this condition.
2.
"PWM_adj <= counter[25:20]" why just 25:20 ? i'ts that mean value in array counter[20],counter[21],....counter[25][/I][/I].
3. "
pwm_width <= pwm_width[5:0]+ pwm_adj;"
in reg/wire declarations pwm adj have 6bit dan pwm width have 7bit, why in just 6 bit in pwm width " + " with pwm adj??
reg [5:0] PWM_adj;
reg [6:0] PWM_width;
5.if in vhdl variable counter is defined as std_logic_vector(26 downto 0) right?