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Discrete vs Analog

Module by: Anders Gjendemsjø

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When comparing analog vs discrete time, we find that there are many similarities. Often we only need to substitute the varible t 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.

Analog

The complex exponential function is defined: xt=ⅇⅈΩt x t Ω t . If Ω(rad/second) is increased the rate of oscillation will increase continuously. The complex exponential function is also periodic for any value of Ω. In figure Figure 1 we have plotted ⅇⅈπt t and ⅇⅈ3πt 3 t (the real parts only). In Figure 1 we see that the red plot, corresponding to a higher value of Ω, has a higher rate of oscillation.

Figure 1: Real parts of complex exponentials.
Figure 1 (analog_complex_exp.png)

Discrete time

The discrete time complex exponential function is defined: xn=ⅇⅈωn x n ω n .

If we increase ω (rad/sample) the rate of oscillation will increase and decrease periodically. The reason is: ⅇⅈω+2πkn=ⅇⅈωnⅇⅈ2πkn=ⅇⅈωn ω 2 k n ω n 2 k n ω n , where n,k∈ℤ n,k .

This implies that the complex exponential with digital angular frequency ω is identical to a complex exponential with ω1=ω+2π ω1 ω 2 , see Figure 2

Figure 2: Two discrete exponentials that are identical
Figure 2 (discrete_complex_exp2.png)
The rate of oscillation will increase until ω=π ω , then it decreases and repeats after 2π. In Figure 3 we see that as we increase the angular frequency towards π the rate of oscillation increases. If you download the Matlab files included at the end of this module you can adjust the parameters and see that the rate of oscillation will decrease when exceeding π (but less than 2π).
Figure 3: Two discrete exponentials with different frequency.
Figure 3 (discrete_complex_exp1.png)

consequence:

We need to consider discrete time exponentials at an (digital angular) frequency interval of 2π only.
Low (digital angular) frequencies will correspond to ω near even multiplies of π. High (digital angular) frequencies will correspond to ω near odd multiplies of π.

Matlab files

complex_exponential.m

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