Thus, the formula and the plot do agree. Choose a delete action Empty this page Remove this page and its subpages. Content is out of sync. You must reload the page to continue. Skip to main content. Available under Creative Commons-ShareAlike 4. Example 4. The magnitude and phase of the periodic pulse sequence's spectrum is shown for positive frequency indices. Signals Represent Information. Analog Signals. Digital Signals.

Structure of Communication Systems. The Fundamental Signal. The Sinusoid Exercise 1.

## Download Conjugate Duality And The Exponential Fourier Spectrum

Exercise 1. Communicating Information with Signals. Introduction Problems. Solutions to Exercises in Chapter 1. Signals and Systems. Complex Numbers. Definitions Exercise 2. Exercise 2. Euler's Formula. Calculating with Complex Numbers Exercise 2. Example 2. Elemental Signals. Complex Exponentials. Real Exponentials. Unit Step. Square Wave. Signal Decomposition Example 2. Discrete-Time Signals.

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Real-and Complex-valued Signals. Unit Sample. Symbolic-valued Signals. Introduction to Systems.

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Cascade Interconnection. Parallel Interconnection. Feedback Interconnection. Simple Systems. Time Reversal Exercise 2.

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Derivative Systems and Integrators. Linear Systems. Time-Invariant Systems. Signals and Systems Problems Problem 2. Problem 2. Solutions to Exercises in Chapter 2. Analog Signal Processing. Ideal Circuit Elements. Ideal and Real-World Circuit Elements. Electric Circuits and Interconnection Laws. Kirchhof's Current Law Exercise 3. Power Dissipation in Resistor Circuits Exercise 3. Exercise 3. Series and Parallel Circuits Exercise 3.

Example 3. Equivalent Circuits: Resistors and Sources Exercise 3. Circuits with Capacitors and Inductors. The Impedance Concept. Time and Frequency Domains Example 3. Power in the Frequency Domain Exercise 3.

Equivalent Circuits: Impedances and Sources Example 3. Transfer Functions Exercise 3. Designing Transfer Functions Example 3. Power Conservation in Circuits. Dependent Sources. Operational Amplifiers. Inverting Amplifier. Active Filters Example 3. The Diode. The duality property is quite useful, but the notation can be tricky. It is sometimes easier to use an alternate statement of the duality principle.

Consider only the case when x t is real the complex case is not much more difficult, but does not pertain to the signals being considered. Represent x t as the sum of an even function and an odd function recall that any function can be represented as the sum of an even part and an odd part. Express the Fourier Transform of x t , substitute the above expression and use Euler's identity for the complex exponential.

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Recall that the product of two odd functions or two even functions is an even function, and the product of an odd and an even function is odd. In other words. Time and Frequency Domains Example 3. Power in the Frequency Domain Exercise 3. Equivalent Circuits: Impedances and Sources Example 3. Transfer Functions Exercise 3.

Designing Transfer Functions Example 3. Power Conservation in Circuits. Dependent Sources. Operational Amplifiers. Inverting Amplifier. Active Filters Example 3. The Diode. Analog Signal Processing Problems Problem 3. Problem 3. Figure 3. Solutions to Exercises in Chapter 3. Frequency Domain. Introduction to the Frequency Domain. Complex Fourier Series Exercise 4.

Classic Fourier Series Exercise 4. A Signal's Spectrum Exercise 4. Fourier Series Approximation of Signals Exercise 4. Encoding Information in the Frequency Domain Exercise 4. Filtering Periodic Signals Example 4. Derivation of the Fourier Transform Example 4. Linear Time Invariant Systems Example 4. Transfer Functions. Commutative Transfer Functions. Modeling the Speech Signal Exercise 4.

Frequency Domain Problems Problem 4. Problem 4. Solutions to Exercises in Chapter 4.

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Digital Signal Processing. Introduction to Digital Signal Processing. Introduction to Computer Organization. Computer Architecture. Representing Numbers Exercise 5. Exercise 5. Computer Arithmetic and Logic Exercise 5. The Sampling Theorem.

Analog-to-Digital Conversion. The Sampling Theorem Exercise 5. Amplitude Quantization Exercise 5. Discrete-Time Signals and Systems. Symbolic Signals. Discrete-Time Systems. DFT: Computational Complexity. Discrete-Time Systems in the Time-Domain. Discrete-Time Systems in the Frequency Domain.

Filtering in the Frequency Domain. Efficiency of Frequency-Domain Filtering. Discrete-Time Filtering of Analog Signals. Digital Signal Processing Problems. Solutions to Exercises in Chapter 5. Information Communication. Types of Communication Channels. Wireline Channels. Wireless Channels.

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## Complex Fourier Series

Line-of-Sight Transmission. The Ionosphere and Communications. Communication with Satellites. Noise and Interference. Channel Models. Baseband Communication. Modulated Communication.

Digital Communication. Binary Phase Shift Keying. Frequency Shift Keying. Digital Communication Receivers. Digital Communication in the Presence of Noise. Digital Communication System Properties.