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Introduction 1-1 Mathematical Representation of Signals 1-2 Mathematical Representation of Systems 1-3 Systems as Building Blocks 1-4 The Next Step Sinusoids 2-1 Tuning Fork Experiment 2-2 Review of Sine and Cosine Functions 2-3 Sinusoidal Signals 2-3.1 Relation of Frequency to Period 2-3.2 Phase and Time Shift 2-4 Sampling and Plotting Sinusoids 2-5 Complex Exponentials and Phasors 2-5.1 Review of Complex Numbers 2-5.2 Complex Exponential Signals 2-5.3 The Rotating Phasor Interpretation 2-5.4 Inverse Euler Formulas Phasor Addition 2-6 Phasor Addition 2-6.1 Addition of Complex Numbers 2-6.2 Phasor Addition Rule 2-6.3 Phasor Addition Rule: Example 2-6.4 MATLAB Demo of Phasors 2-6.5 Summary of the Phasor Addition Rule Physics of the Tuning Fork 2-7.1 Equations from Laws of Physics 2-7.2 General Solution to the Differential Equation 2-7.3 Listening to Tones 2-8 Time Signals: More Than Formulas Summary and Links Problems Spectrum Representation 3-1 The Spectrum of a Sum of Sinusoids 3-1.1 Notation Change 3-1.2 Graphical Plot of the Spectrum 3-1.3 Analysis vs. Synthesis Sinusoidal Amplitude Modulation 3-2.1 Multiplication of Sinusoids 3-2.2 Beat Note Waveform 3-2.3 Amplitude Modulation 3-2.4 AM Spectrum 3-2.5 The Concept of Bandwidth Operations on the Spectrum 3-3.1 Scaling or Adding a Constant 3-3.2 Adding Signals 3-3.3 Time-Shifting x.t/ Multiplies ak by a Complex Exponential 3-3.4 Differentiating x.t/ Multiplies ak by
.j 2nfk/ 3-3.5 Frequency Shifting Periodic Waveforms 3-4.1 Synthetic Vowel 3-4.3 Example of a Non-periodic Signal Fourier Series 3-5.1 Fourier Series: Analysis 3-5.2 Analysis of a Full-Wave Rectified Sine Wave 3-5.3 Spectrum of the FWRS Fourier Series 3-5.3.1 DC Value of Fourier Series 3-5.3.2 Finite Synthesis of a Full-Wave Rectified Sine Time-Frequency Spectrum 3-6.1 Stepped Frequency 3-6.2 Spectrogram Analysis Frequency Modulation: Chirp Signals 3-7.1 Chirp or Linearly Swept Frequency 3-7.2 A Closer Look at Instantaneous Frequency Summary and Links Problems Fourier Series Fourier Series Derivation 4-1.1 Fourier Integral Derivation Examples of Fourier Analysis 4-2.1 The Pulse Wave 4-2.1.1 Spectrum of a Pulse Wave 4-2.1.2 Finite Synthesis of a Pulse Wave 4-2.2 Triangle Wave 4-2.2.1 Spectrum of a Triangle Wave 4-2.2.2 Finite Synthesis of a Triangle Wave 4-2.3 Half-Wave Rectified Sine 4-2.3.1 Finite Synthesis of a Half-Wave Rectified Sine Operations on Fourier Series 4-3.1 Scaling or Adding a Constant 4-3.2 Adding Signals 4-3.3 Time-Scaling 4-3.4 Time-Shifting x.t/ Multiplies ak by a Complex Exponential 4-3.5 Differentiating x.t/ Multiplies ak by
.j!0k/ 4-3.6 Multiply x.t/ by Sinusoid Average Power, Convergence, and Optimality 4-4.1 Derivation of Parseval''s Theorem 4-4.2 Convergence of Fourier Synthesis 4-4.3 Minimum Mean-Square Approximation Pulsed-Doppler Radar Waveform 4-5.1 Measuring Range and Velocity Problems