- #1
Master1022
- 611
- 117
- Homework Statement
- What does it mean when the signal is 'reproduced at the output'?
- Relevant Equations
- Delay
Hi,
I just have a quick question regarding the linear phase property of filters. It might be easier to provide some context before getting to the question, but feel free to skip to the bottom.
Consider a system input as a discrete sequence obtained by sampling at [itex]t = 0, T, . . . , kT[/itex] from an underlying continuous-time function [itex] cos(\omega t + \phi) [/itex]. Thus the input is:
$$x[k] = cos(\omega kT + \phi)$$
and the output of the system, with a linear phase response, is:
$$y[k]=Acos(\omega kT +\phi + \gamma)$$
where [itex] A [/itex] is the gain of the system. If the phase response [itex] \gamma [/itex] is linear, i.e. [itex] \gamma = −\omega k_0 T [/itex], then
$$ y[k] = A cos(\omega(k − k_0 ) T + \phi) $$
That is, the signal waveform has been preserved at the output, with only a delay in time and modification in amplitude. "Notice that when [itex] k_0 [/itex] is not an integer, then the time delay is not integer multiple of the sampling period [itex]T[/itex] . In this case, the input samples are not exactly reproduced at the output."
Question: Why is this final statement in quotation marks true?
I understand that if [itex] k_0 [/itex] is not an integer, then the time delay is not an integer multiple of the sampling period, but I don't understand how this means that the input samples are not exactly reproduced at the output? I feel like this answer should be very simple, but for some reason, this statement doesn't make any sense to me.
Many thanks in advance for any help.
I just have a quick question regarding the linear phase property of filters. It might be easier to provide some context before getting to the question, but feel free to skip to the bottom.
Consider a system input as a discrete sequence obtained by sampling at [itex]t = 0, T, . . . , kT[/itex] from an underlying continuous-time function [itex] cos(\omega t + \phi) [/itex]. Thus the input is:
$$x[k] = cos(\omega kT + \phi)$$
and the output of the system, with a linear phase response, is:
$$y[k]=Acos(\omega kT +\phi + \gamma)$$
where [itex] A [/itex] is the gain of the system. If the phase response [itex] \gamma [/itex] is linear, i.e. [itex] \gamma = −\omega k_0 T [/itex], then
$$ y[k] = A cos(\omega(k − k_0 ) T + \phi) $$
That is, the signal waveform has been preserved at the output, with only a delay in time and modification in amplitude. "Notice that when [itex] k_0 [/itex] is not an integer, then the time delay is not integer multiple of the sampling period [itex]T[/itex] . In this case, the input samples are not exactly reproduced at the output."
Question: Why is this final statement in quotation marks true?
I understand that if [itex] k_0 [/itex] is not an integer, then the time delay is not an integer multiple of the sampling period, but I don't understand how this means that the input samples are not exactly reproduced at the output? I feel like this answer should be very simple, but for some reason, this statement doesn't make any sense to me.
Many thanks in advance for any help.