Example: Spectrum of Square Wave Each line corresponds to � one harmonic frequency. The line magnitude (height) indicates the contribution of that frequency to the signal The line magnitude drops � exponentially, which is not very fast. The very sharp transition in square waves calls for very high frequency sinusoids to synthesize 1 � 30
Negative Frequency? 31
Negative Frequency? 32
Why Frequency Domain Representation of signals? � Shows the frequency composition of the signal � Change the magnitude of any frequency component arbitrarily by a filtering operation � Lowpass -> smoothing, noise removal � Highpass -> edge/transition detection � High emphasis -> edge enhancement � Shift the central frequency by modulation � A core technique for communication , which uses modulation to multiplex many signals into a single composite signal, to be carried over the same physical medium 33
Why Frequency Domain Representation of signals? Typical Filtering applied to x(t): � Lowpass -> smoothing, noise removal � Highpass -> edge/transition detection � Bandpass -> Retain only a certain frequency range 34
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 35
Fourier series of CT periodic signals � Consider the following continuous-time complex exponentials: � T 0 is the period of all of these exponentials and it can be easily verified that the fundamental period is equal to � Any linear combination of is also periodic with period T 0 36
Fourier series of CT periodic signals fundamental components or � the first harmonic components � The corresponding fundamental frequency is ω 0 � Fourier series representation of a periodic signal x(t): (4.1) 37
Fourier series of a CT periodic signal 38
Fourier series of a CT periodic signal 39
Fourier series of a CT periodic signal: Example 4.1 40
Fourier series of a CT periodic signal: Example 4.1 � The Fourier series coefficients are shown in the Figures � Note that the Fourier coefficients are complex numbers in general � Thus one should use two figures to demonstrate them completely: show � real and imaginary parts or � magnitude and angle 41
Fourier series of CT REAL periodic signals � If x(t) is real � This means that 42
Fourier series of CT REAL periodic signals 43
“sinc” Function 44
Fourier series of a CT periodic signal:Example 4.2 45
Fourier series of a CT periodic signal: Example 4.2 � The Fourier series coefficients are shown in Figures for T=4T 1 and T=16T 1 � Note that the Fourier series coefficients for this particular example are real 46
47
Inverse CT Fourier Series: Example: Magnitude and Phase Spectra of the harmonic function X[k] 48
Inverse CT Fourier series: Example � The CT Fourier Series representation of the above cosine ( t ) x F signal X[k] is x F ( t ) is odd � x F ( t ) � The discontinuities make X[k] have significant higher harmonic content 49
Note: Log-Magnitude Frequency Response Plots 50
Note: Log-Magnitude Frequency Response Plots 51
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 52
Effect of Signal Symmetry on CT Fourier Series 53
Effect of Signal Symmetry on CT Fourier Series 54
Effect of Signal Symmetry on CT Fourier Series 55
Effect of Signal Symmetry on CT Fourier Series 56
Effect of Signal Symmetry on CT Fourier Series 57
Effect of Signal Symmetry on CT Fourier Series: Example 58
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 59
Properties of CT Fourier series � The properties are useful in determining the Fourier series or inverse Fourier series � They help to represent a given signal in term of operations (e.g., convolution, differentiation, shift) on another signal for which the Fourier series is known ⇔ � Operations on {x(t)} Operations on {X[k]} � Help find analytical solutions to Fourier Series problems of complex signals � Example: = − → t { ( ) ( 5 ) } FS y t a u t delay and multiplica tion 60
Properties of CT Fourier series � Let x( t ): have a fundamental period T 0x � Let y( t ): have a fundamental period T 0y � Let X[k]=a k and Y[k]=b k � The Fourier Series harmonic functions each using the fundamental period T F as the representation time � In the Fourier series properties which follow: � Assume the two fundamental periods are the same T= T 0x =T 0y (unless otherwise stated) � The following properties can easily been shown using equation (4.5) for Fourier series 61
Properties of CT Fourier series 62
Properties of CT Fourier series 63
Properties of CT Fourier series 64
Properties of CT Fourier series 65
Properties of CT Fourier series 66
Properties of CT Fourier series 67
Properties of CT Fourier series 68
Properties of CT Fourier series 69
Properties of CT Fourier series 70
Properties of CT Fourier series 71
Properties of CT Fourier series: Example 5.1 72
Properties of CT Fourier series: Example 5.2 73
Properties of CT Fourier series: Example 5.2 74
Properties of CT Fourier series: Example 75
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 76
Convergence of the CT Fourier series The Fourier series representation of a periodic signal x(t) � converges to x(t) if the Dirichlet conditions are satisfied Three Dirichlet conditions are as follows: � Over any period, x(t) must be absolutely integrable. 1. For example, the following signal does not satisfy this condition 77
Convergence of the CT Fourier series 2. x(t) must have a finite number of maxima and minima in one period For example, the following signal meets Condition 1, but not Condition 2 78
Convergence of the CT Fourier series 3. x(t) must have a finite number of discontinuities, all of finite size, in one period For example, the following signal violates Condition 3 79
Convergence of the CT Fourier series � Every continuous periodic signal has an FS representation � Many not continuous signals has an FS representation � If a signal x(t) satisfies the Dirichlet conditions and is not continuous, then the Fourier series converges to the midpoint of the left and right limits of x(t) at each discontinuity � Almost all physical periodic signals encountered in engineering practice, including all of the signals with which we will be concerned, satisfy the Dirichlet conditions 80
Convergence of the CT Fourier series: Summary 81
Convergence of the CT Fourier series: Continuous signals 82
Convergence of the CT Fourier series: Discontinuous signals 83
Convergence of the CT Fourier series: Gibb’s phenomenon 84
Convergence of the CT Fourier series: Gibbs Phenomenon 85
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 86
The DT Fourier Series 87
The DT Fourier Series 88
The DT Fourier Series 89
Concept of DT Fourier Series 90
The DT Fourier Series 91
The DT Fourier Series 92
The DT Fourier Series 93
The DT Fourier Series: Example 5.3 94
The DT Fourier Series: Example 5.3 95
The DT Fourier Series: Example 5.3 96
The DT Fourier Series: Example 5.3 97
Outline o Introduction to frequency analysis o Fourier series of CT periodic signals o Signal Symmetry and CT Fourier Series o Properties of CT Fourier series o Convergence of the CT Fourier series o Fourier Series of DT periodic signals o Properties of DT Fourier series o Response of LTI systems to complex exponential o Summary 98
Properties of DT Fourier Series 99
Properties of DT Fourier Series 100
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