Understanding the Use of Variable Notation in the DTFT

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SUMMARY

The discussion clarifies the use of variable notation in the Discrete-Time Fourier Transform (DTFT), specifically the distinction between capital omega (Ω) and the exponential notation e^(jω). Ω represents continuous frequency, which is suitable for the DTFT, while e^(jω) explicitly indicates evaluation on the unit circle, emphasizing frequency response. The z-transform's relevance in Digital Signal Processing (DSP) is also highlighted, as it encompasses the entire complex plane.

PREREQUISITES
  • Understanding of Discrete-Time Fourier Transform (DTFT)
  • Familiarity with complex frequency representation
  • Knowledge of z-transform in Digital Signal Processing (DSP)
  • Basic concepts of frequency response analysis
NEXT STEPS
  • Explore the properties of the Discrete-Time Fourier Transform (DTFT)
  • Study the implications of using capital omega (Ω) in frequency analysis
  • Learn about the z-transform and its applications in DSP
  • Investigate frequency response evaluation using e^(jω)
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Students and professionals in electrical engineering, signal processing practitioners, and anyone interested in understanding frequency analysis in digital systems.

Jammin_James
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Can someone explain to me why sometimes I see the DTFT as functions of capital omegas or e^(jomega).

I'm failing to see the reason.
 
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\Omega usually refers to continuous frequency, which is often appropriate for the DTFT. The z transform is also used heavily in DSP, and it covers the full complex plane. Use of exp(j\omega) makes it very clear that you are evaluating the frequency response (which is on the unit circle).
 

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