Sampling-Rate Conversion: Frequency-Domain Interpretation of Downsampling 

Brilliant work sir

Frequency is scaled by M(2) ,SO it sud be( pie/2 )and not( pie*2)??

very nice work

"Scaling of frequnecy axis by a fator of M" is not true for down sampling I think. When the sentence is true, it is not downsampling , it only change some samples as zero. Downsampling will no change bandwidth if no aliasing happening.

thank u