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When comparing analog vs discrete time, we find that there are many similarities. Often we only need to substitute the variblet with n and integration with summation. Still there are some important differences that we need to know.As the complex exponential signal is truly central to signal processing we will study that in more detail.
The complex exponential function is defined: $x(t)=e^{it}$ . If(rad/second) is increased the rate of oscillation will increase continuously. The complex exponential function is also periodic for any value of. In figure we have plotted $e^{i\pi t}$ and $e^{i\times 3\pi t}$ (the real parts only). In we see that the red plot, corresponding to a higher value of, has a higher rate of oscillation.
The discrete time complex exponential function is defined: $x(n)=e^{in}$ .
If we increase(rad/sample) the rate of oscillation will increase and decrease periodically.The reason is: $e^{i(+2\pi k)n}=e^{in}e^{i\times 2\pi kn}=e^{in}$ , where $n,k\in \mathbb{Z}$ .
This implies that the complex exponential with digital angular frequencyis identical to a complex exponential with ${}_{1}=+2\pi $ , see
The rate of oscillation will increase until $=\pi $ , then it decreases and repeats after 2. In we see that as we increase the angular frequency towardsthe rate of oscillation increases. If you download the Matlab files included at theend of this module you can adjust the parameters and see that the rate of oscillation will decrease when exceeding(but less than 2).Take a look at
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