I. Previously on IET.

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Presentation transcript:

I. Previously on IET

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

Complex Exponential Function Im-Axis ω Re-Axis

The Fourier Transform Representing functions in terms of complex exponentials with different frequencies

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Sine Function) Im-Axis Im-Axis - -ω Re-Axis Re-Axis ω -

The Fourier Transform (Sine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Sine Function) Im-Axis Im-Axis - Re-Axis Re-Axis -ω ω -

The Fourier Transform (Sine Function) Im-Axis -ω Im-Axis + Re-Axis Re-Axis ω +

Fourier Transform of Sinusoids 1/2 1/2 j(1/2) -ω ω -ω ω -j(1/2) Notes A real value for the coefficients in the frequency domain means that the starting point for rotation is on the real axis An Imaginary value for the coefficients in the frequency domain means that the starting point for rotation is on the imaginary axis

Fourier Transform of Real Valued Functions Im-Axis Im-Axis Im-Axis ωn ω1 Re-Axis Re-Axis Re-Axis ω2 Im-Axis Im-Axis Im-Axis -ω2 Re-Axis Re-Axis Re-Axis -ω1 -ωn A real-valued function in time implies that G(-f) = G*(f)