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I chose Ben Horst's function <math>x(t)=\frac{\sin(t)}{t}</math>. The function <math>\sum_{k \in \mathbb{Z}} x(t+2 \pi k) = \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi k)}{t+2 \pi k} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k}</math> is periodic. | I chose Ben Horst's function <math>x(t)=\frac{\sin(t)}{t}</math>. The function <math>\sum_{k \in \mathbb{Z}} x(t+2 \pi k) = \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi k)}{t+2 \pi k} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k}</math> is periodic. | ||
| − | Consider the period <math>P = 2 \pi k</math>. <math>x(t+P) |_{P=2 \pi} = \sum_{k \in \mathbb{Z}} \frac{\sin(t+P)}{(t+P)+2 \pi k}= \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi)}{(t+2 \pi)+2 \pi k}|_{P=2 \pi} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi (k+1)} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k} = x(t) \forall t \in \mathbb{R}</math> | + | Consider the period <math>P = 2 \pi k</math>. <math>x(t+P) |_{P=2 \pi} = \sum_{k \in \mathbb{Z}} \frac{\sin(t+P)}{(t+P)+2 \pi k}= \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi)}{(t+2 \pi)+2 \pi k}|_{P=2 \pi} </math> |
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| + | <math>= \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi (k+1)} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k} = x(t) \forall t \in \mathbb{R}</math> | ||
Revision as of 15:58, 10 September 2008
Part 1
Problem
From the homework:
We have seen in class that sampling a CT periodic signal at regular intervals may or may not yield a periodic DT signal, depending on the sampling frequency. Pick a CT periodic signal that was posted on Rhea as part of homework 1, and create two DT signals, one periodic and one non-periodic, by sampling at different frequencies. Post your answer on a Rhea page.
Solution
Part 2
Problem
From the homework:
One can create a periodic signal by adding together an infinite number of shifted copies of a non-periodic signal periodically (i.e. sum x(t+kT) or x[n+kN] for all integers k). Pick a non-periodic signal that was posted on Rhea as part of homework 1 and create a periodic signal using this method. Post your answer on a Rhea page.
Solution
I chose Ben Horst's function $ x(t)=\frac{\sin(t)}{t} $. The function $ \sum_{k \in \mathbb{Z}} x(t+2 \pi k) = \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi k)}{t+2 \pi k} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k} $ is periodic.
Consider the period $ P = 2 \pi k $. $ x(t+P) |_{P=2 \pi} = \sum_{k \in \mathbb{Z}} \frac{\sin(t+P)}{(t+P)+2 \pi k}= \sum_{k \in \mathbb{Z}} \frac{\sin(t+2 \pi)}{(t+2 \pi)+2 \pi k}|_{P=2 \pi} $
$ = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi (k+1)} = \sum_{k \in \mathbb{Z}} \frac{\sin(t)}{t+2 \pi k} = x(t) \forall t \in \mathbb{R} $
