Preface 

xiv  


1  (6) 

Mathematical Representation of Signals 


2  (2) 

Mathematical Representation of Systems 


4  (1) 


5  (1) 


6  (1) 


7  (29) 


8  (1) 

Review of Sine and Cosine Functions 


9  (2) 


11  (4) 

Relation of Frequency to Period 


12  (1) 

Phase Shift and Time Shift 


13  (2) 

Sampling and Plotting Sinusoids 


15  (2) 

Complex Exponentials and Phasors 


17  (5) 

Review of Complex Numbers 


17  (1) 

Complex Exponential Signals 


18  (1) 

The Rotating Phasor Interpretation 


19  (2) 


21  (1) 


22  (5) 

Addition of Complex Numbers 


23  (1) 


23  (1) 

Phasor Addition Rule: Example 


24  (1) 


25  (1) 

Summary of the Phasor Addition Rule 


26  (1) 

Physics of the Tuning Fork 


27  (2) 

Equations from Laws of Physics 


27  (2) 

General Solution to the Differential Equation 


29  (1) 


29  (1) 

Time Signals: More Than Formulas 


29  (1) 


30  (1) 


31  (5) 


36  (35) 

The Spectrum of a Sum of Sinusoids 


36  (3) 


38  (1) 

Graphical Plot of the Spectrum 


38  (1) 


39  (4) 

Multiplication of Sinusoids 


39  (1) 


40  (1) 


41  (2) 


43  (4) 


44  (1) 

Example of a Nonperiodic Signal 


45  (2) 


47  (3) 


48  (1) 

Fourier Series Derivation 


48  (2) 

Spectrum of the Fourier Series 


50  (1) 

Fourier Analysis of Periodic Signals 


51  (6) 


52  (1) 

DC Value of a Square Wave 


53  (1) 

Spectrum of a Square Wave 


53  (1) 

Synthesis of a Square Wave 


54  (1) 


55  (1) 

Synthesis of a Triangle Wave 


56  (1) 

Convergence of Fourier Synthesis 


57  (1) 


57  (3) 


59  (1) 


59  (1) 

Frequency Modulation: Chirp Signals 


60  (3) 

Chirp or Linearly Swept Frequency 


60  (2) 

A Closer Look at Instantaneous Frequency 


62  (1) 


63  (1) 


64  (7) 


71  (30) 


71  (8) 

Sampling Sinusoidal Signals 


73  (2) 


75  (1) 

Spectrum of a DiscreteTime Signal 


76  (1) 


77  (1) 


78  (1) 

Spectrum View of Sampling and Reconstruction 


79  (5) 

Spectrum of a DiscreteTime Signal Obtained by Sampling 


79  (1) 


79  (2) 

Aliasing Due to UnderSampling 


81  (1) 

Folding Due to UnderSampling 


82  (1) 

Maximum Reconstructed Frequency 


83  (1) 


84  (4) 


87  (1) 

DiscretetoContinuous Conversion 


88  (5) 

Interpolation with Pulses 


88  (1) 

ZeroOrder Hold Interpolation 


89  (1) 


90  (1) 

Cubic Spline Interpolation 


90  (1) 

OverSampling Aids Interpolation 


91  (1) 

Ideal Bandlimited Interpolation 


92  (1) 


93  (1) 


94  (2) 


96  (5) 


101  (29) 


102  (1) 

The RunningAverage Filter 


102  (3) 


105  (6) 

An Illustration of FIR Filtering 


106  (1) 

The Unit Impulse Response 


107  (1) 


107  (1) 

Unit Impulse Response Sequence 


108  (1) 


109  (1) 

Convolution and FIR Filters 


110  (1) 

Computing the Output of a Convolution 


110  (1) 


111  (1) 

Implementation of FIR Filters 


111  (4) 


111  (1) 


112  (1) 


112  (1) 


112  (1) 


113  (1) 


113  (2) 

Internal Hardware Details 


115  (1) 

Linear TimeInvariant (LTI) Systems 


115  (3) 


116  (1) 


117  (1) 


117  (1) 

Convolution and LTI Systems 


118  (4) 

Derivation of the Convolution Sum 


118  (2) 

Some Properties of LTI Systems 


120  (1) 

Convolution as an Operator 


121  (1) 

Commutative Property of Convolution 


121  (1) 

Associative Property of Convolution 


121  (1) 


122  (2) 


124  (2) 


126  (1) 


126  (4) 

Frequency Response of FIR Filters 


130  (33) 

Sinusoidal Response of FIR Systems 


130  (2) 

Superposition and the Frequency Response 


132  (3) 

SteadyState and Transient Response 


135  (2) 

Properties of the Frequency Response 


137  (2) 

Relation to Impulse Response and Difference Equation 


137  (1) 


138  (1) 


138  (1) 

Graphical Representation of the Frequency Response 


139  (4) 


139  (1) 


140  (2) 


142  (1) 


143  (2) 

RunningAverage Filtering 


145  (6) 

Plotting the Frequency Response 


146  (2) 

Cascade of Magnitude and Phase 


148  (1) 

Experiment: Smoothing an Image 


149  (2) 

Filtering Sampled ContinuousTime Signals 


151  (4) 

Example: Lowpass Averager 


152  (2) 


154  (1) 


155  (2) 


157  (6) 


163  (33) 

Definition of the zTransform 


164  (1) 

The zTransform and Linear Systems 


165  (2) 

The zTransform of an FIR Filter 


166  (1) 

Properties of the zTransform 


167  (2) 

The Superposition Property of the zTransform 


168  (1) 

The TimeDelay Property of the zTransform 


168  (1) 

A General zTransform Formula 


169  (1) 

The zTransform as an Operator 


169  (2) 


169  (1) 


170  (1) 

Operator Notation in Block Diagrams 


170  (1) 

Convolution and the zTransform 


171  (4) 


173  (1) 


174  (1) 


175  (1) 

Relationship Between the zDomain and the wDomain 


175  (6) 

The zPlane and the Unit Circle 


176  (1) 

The Zeros and Poles of H (z) 


177  (1) 

Significance of the Zeros of H (z) 


178  (1) 


179  (1) 

Graphical Relation Between z and w 


180  (1) 


181  (5) 

The LPoint RunningSum Filter 


181  (2) 

A Complex Bandpass Filter 


183  (2) 

A Bandpass Filter with Real Coefficients 


185  (1) 

Practical Bandpass Filter Design 


186  (3) 

Properties of LinearPhase Filters 


189  (1) 

The LinearPhase Condition 


189  (1) 

Locations of the Zeros of FIR LinearPhase Systems 


189  (1) 


190  (1) 


191  (5) 


196  (49) 

The General IIR Difference Equation 


197  (1) 


198  (6) 

Linearity and Time Invariance of IIR Filters 


199  (1) 

Impulse Response of a FirstOrder IIR System 


200  (1) 

Response to FiniteLength Inputs 


201  (1) 

Step Response of a FirstOrder Recursive System 


202  (2) 

System Function of an IIR Filter 


204  (6) 

The General FirstOrder Case 


205  (1) 

The System Function and BlockDiagram Structures 


206  (1) 


206  (1) 


207  (1) 

The Transposed Form Structure 


208  (1) 

Relation to the Impulse Response 


209  (1) 


209  (1) 


210  (2) 

Poles or Zeros at the Origin or Infinity 


211  (1) 

Pole Locations and Stability 


211  (1) 

Frequency Response of an IIR Filter 


212  (4) 

Frequency Response using MATLAB 


213  (1) 

ThreeDimensional Plot of a System Function 


214  (2) 


216  (1) 

The Inverse zTransform and Some Applications 


216  (4) 

Revisiting the Step Response of a FirstOrder System 


217  (1) 

A General Procedure for Inverse zTransformation 


218  (2) 

SteadyState Response and Stability 


220  (3) 


223  (8) 

zTransform of SecondOrder Filters 


223  (1) 

Structures for SecondOrder IIR Systems 


224  (1) 


225  (1) 

Impulse Response of a SecondOrder IIR System 


226  (1) 


227  (1) 


228  (3) 

Frequency Response of SecondOrder IIR Filter 


231  (5) 

Frequency Response via MATLAB 


232  (1) 


232  (1) 

ThreeDimensional Plot of System Functions 


233  (3) 

Example of an IIR Lowpass Filter 


236  (1) 


237  (1) 


238  (7) 

ContinuousTime Signals and LTI Systems 


245  (40) 


246  (2) 

TwoSided InfiniteLength Signals 


246  (1) 


247  (1) 


248  (1) 


248  (6) 

Sampling Property of the Impulse 


250  (2) 


252  (1) 


252  (1) 

Derivative of the Unit Step 


252  (2) 


254  (1) 

Some Basic ContinuousTime Systems 


254  (1) 


255  (1) 

Analogous DiscreteTime Systems 


255  (1) 

Linear TimeInvariant Systems 


255  (5) 


256  (1) 


256  (1) 


257  (2) 

Properties of Convolution 


259  (1) 

Impulse Responses of Basic LTI Systems 


260  (1) 


260  (1) 


261  (1) 


261  (1) 


261  (2) 

Evaluating Convolution Integrals 


263  (7) 


263  (4) 

Evaluation of Discrete Convolution 


267  (1) 


268  (1) 

Very Narrow Square Pulse Input 


269  (1) 

Discussion of Convolution Examples 


270  (1) 

Properties of LTI Systems 


270  (6) 

Cascade and Parallel Combinations 


270  (2) 

Differentiation and Integration of Convolution 


272  (1) 


273  (3) 

Using Convolution to Remove Multipath Distortion 


276  (2) 


278  (1) 


279  (6) 


285  (22) 

The Frequency Response Function for LTI Systems 


285  (4) 

Plotting the Frequency Response 


287  (1) 


288  (1) 

Magnitude and Phase Changes 


288  (1) 

Response to Real Sinusoidal Signals 


289  (6) 


290  (1) 


290  (3) 

Response to a General Sum of Sinusoids 


293  (1) 


294  (1) 


295  (3) 


295  (1) 


296  (1) 


297  (1) 


297  (1) 

Application of Ideal Filters 


298  (2) 

TimeDomain or FrequencyDomain? 


300  (1) 


301  (1) 


302  (5) 

ContinuousTime Fourier Transform 


307  (39) 

Definition of the Fourier Transform 


308  (2) 

Fourier Transform and the Spectrum 


310  (2) 

Limit of the Fourier Series 


310  (2) 

Existence and Convergence of the Fourier Transform 


312  (1) 

Examples of Fourier Transform Pairs 


313  (9) 

RightSided Real Exponential Signals 


313  (1) 


314  (1) 

Rectangular Pulse Signals 


314  (2) 


316  (1) 

Impulse in Time or Frequency 


317  (1) 


318  (1) 


319  (3) 

Properties of Fourier Transform Pairs 


322  (4) 


322  (2) 

Symmetry Properties of Fourier Transform Pairs 


324  (2) 


326  (6) 


326  (1) 

Fourier Transform of a Convolution 


327  (1) 

Examples of the Use of the Convolution Property 


328  (1) 

Convolution of Two Bandlimited Functions 


328  (1) 

Product of Two Sinc Functions 


329  (1) 

Partial Fraction Expansions 


330  (2) 


332  (3) 


332  (1) 


333  (1) 

Systems Described by Differential Equations 


334  (1) 

The Multiplication Property 


335  (2) 

The General Signal Multiplication Property 


335  (1) 

The Frequency Shifting Property 


336  (1) 

Table of Fourier Transform Properties and Pairs 


337  (1) 

Using the Fourier Transform for Multipath Analysis 


337  (4) 


341  (1) 


342  (4) 

Filtering, Modulation, and Sampling 


346  (43) 

Linear TimeInvariant Systems 


346  (12) 

Cascade and Parallel Configurations 


347  (1) 


348  (3) 

Frequency Selective Filters 


351  (1) 


351  (1) 

Other Ideal Frequency Selective Filters 


352  (1) 

Example of Filtering in the FrequencyDomain 


353  (2) 

Compensation for the Effect of an LTI Filter 


355  (3) 

Sinewave Amplitude Modulation 


358  (10) 

DoubleSideband Amplitude Modulation 


358  (4) 

DSBAM with Transmitted Carrier (DSBAMTC) 


362  (4) 

Frequency Division Multiplexing 


366  (2) 

Sampling and Reconstruction 


368  (12) 

The Sampling Theorem and Aliasing 


368  (2) 

Bandlimited Signal Reconstruction 


370  (2) 

Bandlimited Interpolation 


372  (1) 

Ideal CtoD and DtoC Converters 


373  (2) 

The DiscreteTime Fourier Transform 


375  (1) 


376  (1) 

DiscreteTime Filtering of ContinuousTime Signals 


377  (3) 


380  (1) 


381  (8) 


389  (38) 


390  (1) 

Too Many Fourier Transforms? 


391  (2) 

Relation of the DTFT to the CTFT 


392  (1) 

Relation of the DFT to the DTFT 


393  (1) 

Relation of the DFT to the CTFT 


393  (1) 


393  (2) 

Analysis of a Sum of Sinusoids 


395  (4) 

DTFT of a Windowed Sinusoid 


398  (1) 

Discrete Fourier Transform 


399  (6) 


400  (1) 

Summary of the DFT Representation 


401  (1) 

The Fast Fourier Transform (FFT) 


402  (1) 

Negative Frequencies and the DFT 


402  (1) 


403  (2) 

Spectrum Analysis of FiniteLength Signals 


405  (2) 

Spectrum Analysis of Periodic Signals 


407  (1) 


408  (12) 


409  (1) 


410  (1) 

Spectrogram of a Sampled Periodic Signal 


410  (1) 

Resolution of the Spectrogram 


411  (1) 


412  (1) 

Spectrogram of a Musical Scale 


413  (2) 

Spectrogram of a Speech Signal 


415  (3) 


418  (2) 

The Fast Fourier Transform (FFT) 


420  (3) 


420  (1) 


421  (2) 


423  (1) 


424  (3) 


427  (16) 


428  (1) 

Notation for Complex Numbers 


428  (3) 


428  (1) 


429  (1) 

Conversion: Rectangular and Polar 


430  (1) 

Difficulty in Second or Third Quadrant 


431  (1) 


431  (1) 


432  (1) 

Algebraic Rules for Complex Numbers 


432  (2) 


434  (1) 

Geometric Views of Complex Operations 


434  (4) 

Geometric View of Addition 


435  (1) 

Geometric View of Subtraction 


436  (1) 

Geometric View of Multiplication 


437  (1) 

Geometric View of Division 


437  (1) 

Geometric View of the Inverse, z1 


437  (1) 

Geometric View of the Conjugate, z* 


438  (1) 


438  (3) 


439  (1) 

Procedure for Finding Multiple Roots 


440  (1) 


441  (1) 


441  (2) 


443  (12) 


444  (1) 

Matrix Operations and Variables 


444  (2) 


445  (1) 

Matrix and Array Operations 


445  (1) 

A Review of Matrix Multiplication 


445  (1) 

Pointwise Array Operations 


446  (1) 


446  (1) 


447  (1) 


447  (1) 

Printing and Saving Graphics 


447  (1) 


447  (1) 

MATLAB Builtin Functions 


448  (1) 


448  (1) 


448  (1) 

Writing a MATLAB Function 


448  (3) 


449  (2) 

Debugging a MATLAB Mfile 


451  (1) 


451  (4) 


452  (1) 

Repeating Rows or Columns 


452  (1) 

Vectorizing Logical Operations 


452  (1) 


453  (1) 


453  (1) 


454  (1) 


454  (1) 


455  (23) 


457  (6) 


457  (1) 


457  (1) 


457  (1) 


458  (1) 


458  (1) 


459  (1) 


459  (1) 


460  (1) 

Laboratory: Manipulating Sinusoids with MATLAB 


460  (1) 


461  (1) 


461  (1) 


461  (2) 

Encoding and Decoding TouchTone Signals 


463  (10) 


463  (1) 


463  (1) 

Background: Telephone TouchTone Dialing 


463  (1) 


464  (1) 


464  (1) 


464  (1) 


465  (1) 


465  (1) 


465  (1) 


465  (1) 


466  (1) 

Simple Bandpass Filter Design 


467  (1) 


468  (1) 

Filter Bank Design: dtmfdesign.m 


468  (1) 

A Scoring Function: dtmfscore.m 


469  (1) 

DTMF Decode Function: dtmfrun.m 


470  (1) 


471  (1) 


471  (1) 


472  (1) 


473  (5) 


473  (1) 


473  (1) 

DiscreteTime Convolution Demo 


473  (1) 

ContinuousTime Convolution Demo 


474  (1) 


475  (1) 

ContinuousTime Convolution GUI 


475  (1) 


475  (1) 


475  (1) 

ContinuousTime Convolution 


475  (1) 

ContinuousTime Convolution Again 


476  (1) 

DiscreteTime Convolution 


476  (2) 


478  (4) 
Index 

482  