Time scaing in discrete time variable?

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SUMMARY

The discussion centers on the relationship between the Discrete Time Fourier Transform (DTFT) of a scaled signal and its original signal. Specifically, for the case where \( y(n) = x(a \cdot n) \) with \( a \) as an integer, the DTFT \( Y(e^{j\omega}) \) can be defined as \( (1/a) \sum_{m=0}^{a-1} X(e^{j(\omega + 2\pi m)/a}) \). However, for the case \( y(n) = x(n/a) \), the presence of zeros in the signal raises questions about the feasibility of defining \( Y(e^{j\omega}) \) in terms of \( X(e^{j\omega}) \).

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  • Understanding of Discrete Time Fourier Transform (DTFT)
  • Knowledge of signal scaling in discrete time
  • Familiarity with integer and non-integer scaling factors
  • Basic concepts of signal processing
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  • Research the implications of scaling signals in the context of DTFT
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ratn_kumbh
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I wanted to know , if x(n) has DTFT X(e^jw)
then can we define Y(e^jw) in terms of X(e^jw)?
where Y(e^jw)is DTFT of y(n)=x(a*n)or y(n)=x(n/a)
. Because in these cases terms of x(n) are either missed or '0' is padded up, so i think it won't be possible to define Y(e^jw) in terms of X(e^jw). can anybody tell i m right or not?
 
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OK, i got the answer for y(n)=x(a*n), where a is an integer. Y(e^j[tex]\omega[/tex]) can be defined.

it is (1/a)*[tex]\sum[/tex]X(exp(j([tex]\omega[/tex]+2[tex]\pi[/tex]m)/a)) where m varies from 0 to a-1.

But can anybody please tell; is it possible for y(n)=x(n/a) to define Y(e^j[tex]\omega[/tex]). i am getting confused becoz of 0's which come in y(n) in this case.
 
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