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Read : Ch. 17 in Electric Circuits, 9 th Edition by Nilsson

Lecture #22 EGR 261 – Signals and Systems. Read : Ch. 17 in Electric Circuits, 9 th Edition by Nilsson Ch. 7 in Linear Systems and Signals, 2 nd Edition by Lathi. Fourier Transforms

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Read : Ch. 17 in Electric Circuits, 9 th Edition by Nilsson

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  1. Lecture #22 EGR 261 – Signals and Systems Read: Ch. 17 in Electric Circuits, 9th Edition by Nilsson Ch. 7 in Linear Systems and Signals, 2nd Edition by Lathi Fourier Transforms Recall from last class that the following integral definitions were developed for the Fourier transform and the inverse Fourier transform: Fourier transform Inverse Fourier transform Example (Assessing Objective 17.1 in Electric Circuits, by Nilsson) Use the defining integral to find the Fourier transform of the following function.

  2. Lecture #22 EGR 261 – Signals and Systems Example(continued) Also find |F(w)| and graph |F(w)| versus w using Excel if A = 10 and  = 4.

  3. Lecture #22 EGR 261 – Signals and Systems Example (Assessing Objective 17.2 in Electric Circuits, by Nilsson) Find f(t) if the Fourier transform of f(t) is specified as follows. Sketch f(t) and F(w).

  4. Lecture #22 EGR 261 – Signals and Systems Example Find the Fourier transform of f(t) = (t). Sketch f(t) and F(w). Example Find the inverse Fourier transform of F(w) = (w). Sketch f(t) and F(w).

  5. Lecture #22 EGR 261 – Signals and Systems Fourier Transform Properties As with Laplace transforms, we can often find the Fourier transform of a complex function by applying properties to Fourier transforms of known functions. Discuss each property. Reference: Table 7.2 in Linear Systems and Signals, 2nd Edition, by Lathi.

  6. Lecture #22 EGR 261 – Signals and Systems Table of Fourier Transforms In order to effectively use the Fourier transform properties just introduced, we must begin with a table of known Fourier transforms. Note that we have derived several of the transforms in the table shown below. Reference: Table 7.1 in Linear Systems and Signals, 2nd Edition, by Lathi.

  7. Lecture #22 EGR 261 – Signals and Systems Fourier Transform Properties Scaling (homogeneity) property: If f(t) ↔F(w) then Proof: kf(t) ↔ kF(w) Example: Find F(w) if f(t) = 20cos(500t). Also sketch F(w).

  8. Lecture #22 EGR 261 – Signals and Systems Fourier Transform Properties Additivity (superposition) property: If f1(t) ↔F1(w) and f2(t) ↔F2(w) then f1(t)+ f2(t) ↔ F1(w) + F2(w) Example: Find F(w) if f(t) = 10 + 10cos(25t). Also sketch F(w).

  9. Lecture #22 EGR 261 – Signals and Systems Example: Find V(w) for v(t) - a Fourier series. Also sketch |V(w)|.

  10. Lecture #22 EGR 261 – Signals and Systems Fourier Transform Properties Complex Conjugation property: If f(t) ↔F(w) then f*(t) ↔ F*(-w) Proof: Example: If f(t) = ejwot, find F(w) from the table of Fourier transforms. Also find the Fourier transform of f*(t). Example: Sketch F(w) for ejwot and e-jwot . Using Euler’s identity for cos(wot), sketch its Fourier transform also.

  11. Illustration: Fourier transform of a gated pulse x(t) is real-valued X(w) has odd symmetry |X(w)| has even symmetry Lecture #22 EGR 261 – Signals and Systems Application of Complex Conjugation Property: Suppose that f(t) is a real-valued function. If f(t) ↔F(w) and f*(t) ↔ F*(-w) Then since f(t) = f*(t) for a real-valued function, it follows that F(w) = F*(-w). And if F(w) = F*(-w), then |F(w)| = |F*(-w)| = |F(-w)|, so |F(w)| has even symmetry and if F(w) = F*(-w), then F(w) = F*(-w) = -F(-w), so F(w) has odd symmetry Summary: For real-valued functions, |F(w)| has even symmetry and F(w) has odd symmetry.

  12. Lecture #22 EGR 261 – Signals and Systems Fourier Transform Properties F(t) ↔ 2πf(-w) Duality: If f(t) ↔F(w) then Notation: f(-w) = f(t) where t has been replaced by -w F(t) = F(w) where w has been replaced by t Meaning: If w is replaced by t in a known F(w), then the Fourier transform can be found by replacing t with –w and multiplying by 2π. Note: This almost doubles our table of known transforms! Explain.

  13. x(t) X(w) X(t) 2πx(-w) Lecture #22 EGR 261 – Signals and Systems Example using Duality Property: Duality: If f(t) ↔F(w) then F(t) ↔ 2πf(-w) Illustration:

  14. Lecture #22 EGR 261 – Signals and Systems Example: Use the duality property to find the transform of the following functions. A) B) There is nothing like this in our table under f(t), but there is under F(w), so use the duality property.

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