Text Slecture - Rhea

Outline

• Introduction
• Useful Background
• DTFT Example of a Cosine Sampled Above the Nyquist Rate
• DTFT Example of a Cosine Sampled Below the Nyquist Rate
• Conclusion
• References

Introduction

In this Slecture, I will walk you through taking the DTFT of a pure frequency sampled above and below the Nyquist Rate. Then I will compare the differences between them.

Useful Background

Nyquist Condition: $ f_s = 2f_{max} $
DTFT of a Cosine: $ x_d[n] = cos(2\pi nT){\leftrightarrow}X(\omega) = \pi(\delta(\omega-\omega_o) + \delta(\omega+\omega_o)){ ,for\ } \omega \in [-\pi,\pi] $

The DTFT of a sampled signal is periodic with 2π.

DTFT of a Cosine Sampled Above the Nyquist Rate

x(t) = cos(2π * 330t)

Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is:
 f_s = 2 * f_max = 2 * (330Hz) = 660Hz. Let’s sample at 990Hz. $ \begin{align} \\ x_d[n] & = x(n*\frac{1}{990Hz})\\ & = cos(2\pi n *\frac{330}{990}) = \frac{e^{j2\pi n \frac{330}{990}} + e^{-j2\pi n \frac{330}{990}}}{2}\\ & = cos(\frac{2\pi n}{3}) \end{align} $
Because $ \left | \frac{2\pi}{3}\right | < \pi $, there is no aliasing occurring in the DTFT, and it can be written as follows: $ \begin{align} \\ X(\omega) & = \frac{1}{2}(2\pi\delta(\omega - 2\pi \frac{330}{990}) + 2\pi\delta(\omega + 2\pi \frac{330}{990})) , \ \omega \in\ [-\pi,\pi]\\ & = \frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330)) , \ \omega \in\ [-\pi,\pi]\\ & = rep_{2\pi}(\frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330))), \forall \omega \end{align} $

DTFT Of a Cosine Sampled Below the Nyquist Rate

x(t) = cos(2π * 330t)

Now let’s sample this pure cosine at a frequency below the Nyquist Rate. From above, the Nyquist Rate is 660Hz. Let’s sample at 550Hz.

$ \begin{align} \\ x_d[n] & = x(n*\frac{1}{550Hz})\\ & = cos(2\pi n *\frac{330}{550}) = \frac{e^{j2\pi n \frac{330}{550}} + e^{-j2\pi n \frac{330}{550}}}{2}\\ \\ \end{align} $

Because $ \pi < \frac{2\pi 330}{550} < 2\pi $, aliasing occurs in the DTFT. The DTFT should be calculated with ω ∈ [ − π, π], so we will use the periodicity of cosine to shift $ x_d[n] $ into an appropriate range.
>$ \begin{align}\\ x_{d}[n] & = cos(2\pi n*\frac{330}{550})\\ & = cos(2\pi n*\frac{330}{550} - 2\pi n)\\ & = cos(2\pi n*(\frac{330}{550} - \frac{550}{550}))\\ & = cos(2\pi n*(\frac{-220}{550}))\\ & = cos(2\pi n*\frac{220Hz}{550Hz}) \end{align} $

Now that the argument of the cosine $ \left | 2\pi \frac{220}{550}\right | < \pi $, we can take the DTFT of x_d[n], and the initial value will fall into a desired range for ω.

$ \begin{align} X(\omega) & = \frac{1}{2}(2\pi\delta(\omega - 2\pi \frac{220}{550}) + 2\pi\delta(\omega + 2\pi \frac{220}{550})) , \ \omega \in\ [-\pi,\pi]\\ & = \frac{550}{2}(\delta(\frac{550}{2\pi}\omega - 220) + \delta(\frac{550}{2\pi}\omega + 220)) , \ \omega \in\ [-\pi,\pi]\\ & = rep_{2\pi}(\frac{550}{2}(\delta(\frac{550}{2\pi}\omega - 220) + \delta(\frac{550}{2\pi}\omega + 220))), \forall \omega \end{align} $

Conclusion

The DTFT of a sampled signal is always periodic with $ 2\pi $. So even though the DTFT of a signal sampled below Nyquist may initially not fall within $ [-\pi,\pi] $, it can be extrapolated to the window you are interested in. In my derivation, I chose to shift the cosine before the DTFT. Looking at Figure 3, you can see the comparison between a cosine sampled above and below the Nyquist Rate. The cosine sampled below the Nyquist Rate exhibits aliasing. The aliased signal has a decreased magnitude compared to the original. The aliased signal also is at a different frequency.

References

[1] Mireille Boutin, "ECE 438 Digital Signal Processing with Applications," Purdue University. September 9, 2014.

Questions and comments

If you have any questions, comments, etc. please post them on this page

Back to ECE438 slectures, Fall 2014