Time scaing in discrete time variable?

[ratn_kumbh]

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?

 

[ratn_kumbh]

continue

OK, i got the answer for y(n)=x(a*n), where a is an integer. Y(e^j$$\omega$$) can be defined.

it is (1/a)*$$\sum$$X(exp(j($$\omega$$+2$$\pi$$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$$\omega$$). i am getting confused becoz of 0's which come in y(n) in this case.

 

[tacman]

Time scaling in discrete time variables refers to the manipulation of a signal's time axis by a scaling factor. This can be achieved by multiplying the signal's time index by a constant, a. The resulting signal will have the same values as the original signal, but at different time instances.

In terms of the DTFT, time scaling can be represented as a change in the frequency variable, w. Specifically, if x(n) has a DTFT X(e^jw), then y(n) = x(a*n) will have a DTFT Y(e^j(a*w)). This means that the frequency components of the original signal have been multiplied by a factor of a.

In the case of y(n) = x(n/a), the frequency components will be divided by a, which can result in aliasing if the original signal is not bandlimited. In this case, it may not be possible to define Y(e^jw) in terms of X(e^jw) because the frequency components of the original signal may not be recoverable.

Similarly, if terms of x(n) are missed or '0' is padded up, the resulting signal may not have a well-defined DTFT and therefore it may not be possible to define Y(e^jw) in terms of X(e^jw).

In conclusion, it is not always possible to define Y(e^jw) in terms of X(e^jw) when time scaling is applied to a discrete time variable. This depends on the specific scaling factor and the characteristics of the original signal.