# upsampling for a signal

We have a signal $$s(t)$$.

1. If we do an upsampling, does the signal duration increase?
2. What is the point of upsampling if the signal time increases?
3. Can we do an upsampling if we don't use a shaping filter?

• If your signal $$s[n]$$ is sampled at a rate $$f_s$$ Hz, and it is $$N$$ samples long, then the total duration of the signal is $$\frac{N}{f_s}$$ seconds.
• Say you upsample the signal by a factor $$M$$, yielding a new signal $$s_u[n]$$. It will be $$MN$$ samples long, but it will be sampled at a new rate $$Mf_s$$. Its duration is therefore $$\frac{MN}{Mf_s} = \frac{N}{f_s}$$. You can see that the signal duration is the same as $$s[n]$$.