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I'm having some difficulty in going through the z-transform of a sequence that is "on" every other sample. The sequence is $$x(n) = na^{|n|/2},$$ when $n$ is an even integer, and 0 otherwise. I have understanding how to go about performing the z-transform itself and using the properties, but with this sequence alternating is throwing me for a loop. Any help would be appreciated.

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I would approach this by first thinking of this as an upsampled version of a different signal, x(m) where m=2n, which has values at every integer.

$$x(m) = ma^{|m|/2} = 2na^{|n|}$$

So x(m=2) has the same value as x(n=4) since m=2n and x(m) is non-alternating

Then all you have to do is find the z transform of the non-alternating sequence x(m), G(z) and use the upsampling property of the z-transform(http://web.stanford.edu/class/archive/ee/ee264/ee264.1052/upsampling.pdf) to know that if:

$$x(m) \rightarrow G(z)$$ $$x(n) \rightarrow F(z)=G(z^{2})$$

If my explanation isn't sufficient let me know and I'll try to help more.

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  • $\begingroup$ Thanks Bootstrap. That makes sense. Little bit of confusion on the unsampled version you mention. Since the sampled version is already the x(n)you mention, I would do the substitution to get it into a similar unsampled version, where $$m = n/2. Does that seem right? $\endgroup$ – Christopher Mark Aug 28 '17 at 23:14
  • $\begingroup$ I think so. I honestly get pretty confused with the notation with upsampling and downsampling so I try not to worry to much about it and just make sure I have it straight inside my head. For example I know the downsampled version should have the same values as the upsampled version just at different indicies. I probably should have named the downsampled version y(m), set up the equations so for example y(2)=x(4). Once that is correct just take the Z transform of y(m) and then replace z with z^2 $\endgroup$ – Bootstrap Aug 28 '17 at 23:23
  • $\begingroup$ I think I'm following you correctly. This would be upsampling in this case. So I would take the z-transform of the x(n) you provided $$ x(n) = 2na^abs(n) $$, if I'm following correctly. Then, replace the z with z^2. The next step I see is trying to reduce that series into a simple form of a geometric series, however, I'm not sure what the reduced form would be. $\endgroup$ – Christopher Mark Aug 28 '17 at 23:34
  • $\begingroup$ The key is to realize you can split the absolute value into two parts: $$ x(n) = 2na^{|n|}= 2[na^{-n}u(-n)+ na^{n}u(n)]$$ where u(n) is a right facing unit step and u(-n) is a left facing unitstep. You then can look up the individual parts in a z-transform table $\endgroup$ – Bootstrap Aug 28 '17 at 23:56
  • $\begingroup$ Makes sense. Rather than using a table, I'm curious how one would go about reducing it through means of the definition of a geometric series for the causal and non-causal portions. $\endgroup$ – Christopher Mark Aug 29 '17 at 0:21

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