Text Slecture - Rhea

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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 = 2 * f_max
DTFT of a Cosine:  x_d[n] = cos(2πnT) → X(ω) = π(δ(ω − ω_o) + δ(ω + ω_o)), for ω  ∈  [ − π, π]
The DTFT of a sampled signal is periodic with 2π.

DTFT of a Cosine Sampled Above the Nyquist Rate

For our original pure frequency, let’s choose the E below middle C. The E occurs at 330Hz.

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. Back to ECE438, Fall 2014