3.0 Fourier Series Representation of Periodic Signals

3.0 Fourier Series Representation of Periodic Signals
3.1 Exponential/Sinusoidal Signals as Building Blocks for Many Signals

Time/Frequency Domain Basis Sets
Time Domain 𝑛 𝑥[𝑛] 𝑥(𝑡) 𝑡 {𝛿 𝑡−𝜏 , −∞<𝜏<∞} {𝛿 𝑛−𝑘 , 𝑘=0,±1,±2, ⋯} Frequency Domain { 𝑒 𝑗𝜔𝑡 , −∞<𝜔<∞} { 𝑒 𝑗𝜔𝑛 ,𝜔∈ 0, 2𝜋 } 𝐴 = 𝑘 𝑎 𝑘 𝑣 𝑘 (合成) 𝑎 𝑗 = 𝐴 ⋅ 𝑣 𝑗 分析 𝑡, 𝑛

Signal Analysis (P.32 of 1.0) 𝑅𝑒{ }= cos 𝜔 𝑘 𝑡

Response of A Linear Time-invariant System to An Exponential Signal
Initial Observation time-invariant scaling property 0 t if the input has a single frequency component, the output will be exactly the same single frequency component, except scaled by a constant

Input/Output Relationship
𝑦(𝑡) 𝑥(𝑡) Time Domain 𝛿(𝑡) ℎ(𝑡) 𝑡 Frequency Domain 𝜔 𝜔 𝜔 1 𝜔 3 𝜔 5 𝜔 1 𝜔 3 𝜔 5

More Complete Analysis continuous-time matrix vectors 𝐴 𝑥=𝑦 𝐴 𝜈=𝜆𝜈 eigen value eigen vector

More Complete Analysis continuous-time Transfer Function Frequency Response : eigenfunction of any linear time-invariant system : eigenvalue associated with the eigenfunction est

More Complete Analysis discrete-time Transfer Function Frequency Response eigenfunction, eigenvalue

System Characterization
Superposition Property continuous-time discrete-time each frequency component never split to other frequency components, no convolution involved desirable to decompose signals in terms of such eigenfunctions

3.2 Fourier Series Representation of Continuous-time Periodic Signals
Harmonically related complex exponentials T: fundamental period all with period T

Harmonically Related Exponentials for Periodic Signals
𝑉= 𝑥(𝑡) 𝑥(𝑡) perodic fundamental period [𝑛] =𝑇(𝑁) T (𝑁) 𝑡, 𝑛 𝑎 4 𝑡, 𝑛 𝜔 0 = 2𝜋 𝑇 𝑎 1 𝑎 3 𝑎 5 (𝑁) 𝑎 0 𝑘=1 𝑘 𝑡, 𝑛 𝜔 𝑘=2 𝜔 0 3𝜔 0 5𝜔 0 𝑡, 𝑛 2𝜔 0 4𝜔 0 𝑘=3 All with period T: integer multiples of ω0 Discrete in frequency domain (𝑁)

Fourier Series Representation
: j-th harmonic components real

Real Signals For orthogonal basis: (unique representation)
𝑘 𝑎 𝑘 𝑣 𝑘 = 𝑘 𝑏 𝑘 𝑣 𝑘 𝑘 ( 𝑎 𝑘 − 𝑏 𝑘 ) 𝑣 𝑘 =0⇒ 𝑎 𝑘 = 𝑏 𝑘 (unique representation)

Determination of ak Fourier series coefficients dc component

𝐴 ⋅ 𝑣 𝑛 =( 𝑘 𝑎 𝑘 𝑣 𝑘 )⋅ 𝑣 𝑛 𝑣 𝑘 ⋅ 𝑣 𝑛 = 𝑇, 𝑘=𝑛 0,𝑘≠𝑛 Not unit vector
𝐴 ⋅ 𝑣 𝑛 =( 𝑘 𝑎 𝑘 𝑣 𝑘 )⋅ 𝑣 𝑛 𝑣 𝑘 ⋅ 𝑣 𝑛 = 𝑇, 𝑘=𝑛 0,𝑘≠𝑛 Not unit vector orthogonal 𝐴 ⋅ 𝑣 𝑛 =𝑇 𝑎 𝑛 𝑎 𝑛 = 1 𝑇 ( 𝐴 ⋅ 𝑣 𝑛 ) (分析)

Vector Space Interpretation vector space could be a vector space some special signals (not concerned here) may need to be excluded

Vector Space Interpretation orthonormal basis is a set of orthonormal basis expanding a vector space of periodic signals with period T

Vector Space Interpretation Fourier Series

Completeness Issue Question: Can all signals with period T be represented this way? Almost all signals concerned here can, with exceptions very often not important

Convergence Issue consider a finite series It can be shown ak obtained above is exactly the value needed even for a finite series

Truncated Dimensions 𝑎 1 ′ = 𝑎 1 , 𝑎 2 ′ = 𝑎 2 ,
𝑎 1 ′ = 𝑎 1 , 𝑎 2 ′ = 𝑎 2 , for orthogonal 𝑖 , 𝑗 , 𝑘 All truncated dimensions are orthogonal to the subspace of dimensions kept.

Convergence Issue It can be shown

Gibbs Phenomenon the partial sum in the vicinity of the discontinuity exhibit ripples whose amplitude does not seem to decrease with increasing N See Fig. 3.9, p.201 of text

Convergence Issue x(t) has no discontinuities Fourier series converges to x(t) at every t x(t) has finite number of discontinuities in each period Fourier series converges to x(t) at every t except at the discontinuity points, at which the series converges to the average value for both sides All basis signals are continuous, so converge to average values

Convergence Issue Dirichlet’s condition for Fourier series expansion (1) absolutely integrable,  (2) finite number of maxima & minima in a period (3) finite number of discontinuities in a period

3.3 Properties of Fourier Series
Linearity 𝑥 =( 𝑎 1 , 𝑎 2 , 𝑎 3 ,⋯) 𝑦 =( 𝑏 1 , 𝑏 2 , 𝑏 3 ,⋯) 𝐴 𝑥 +𝐵 𝑦 =(𝐴 𝑎 1 +𝐵 𝑏 1 ,𝐴 𝑎 2 +𝐵 𝑏 2 ,⋯)

Time Shift phase shift linear in frequency with amplitude unchanged
𝑎 𝑘 𝑒 𝑗𝑘 𝜔 0 (𝑡− 𝑡 0 ) = 𝑒 −𝑗𝑘 𝜔 0 𝑡 0 𝑎 𝑘 𝑒 𝑗𝑘 𝜔 0 𝑡 𝑡 0 𝑥(𝑡−𝑡 0 ) 𝑥(𝑡) 𝜔 0 2𝜔 0 3𝜔 0 𝑘𝜔 0 𝑡 0 𝜋 2𝜋 3𝜋

unique representation for orthogonal basis
Time Reversal the effect of sign change for x(t) and ak are identical unique representation for orthogonal basis

Time Scaling positive real number
periodic with period T/α and fundamental frequency αω0 ak unchanged, but x(αt) and each harmonic component are different

Multiplication (⋯, 𝑎 0 𝑏 3 + 𝑎 1 𝑏 2 + 𝑎 2 𝑏 1 + 𝑎 3 𝑏 0 ,⋯) 𝑒 𝑗3 𝜔 0 𝑡 𝑑 3 = 𝑗 𝑎 𝑗 𝑏 3−𝑗

Conjugation unique representation

Differentiation

Parseval’s Relation 𝐴 2 = 𝑘 𝑎 𝑘 2 but 𝑣 𝑖 ⋅ 𝑣 𝑗 =𝑇 𝛿 𝑖𝑗
𝐴 2 = 𝑘 𝑎 𝑘 2 but 𝑣 𝑖 ⋅ 𝑣 𝑗 =𝑇 𝛿 𝑖𝑗 total average power in a period T average power in the k-th harmonic component in a period T

3.4 Fourier Series Representation of Discrete-time Periodic Signals
Harmonically related signal sets all with period only N distinct signals in the set

Harmonically Related Exponentials for Periodic Signals
(P. 11 of 3.0) 𝑉= 𝑥(𝑡) 𝑥(𝑡) perodic fundamental period [𝑛] =𝑇(𝑁) T (𝑁) 𝑡, 𝑛 𝑎 4 𝑡, 𝑛 𝜔 0 = 2𝜋 𝑇 𝑎 1 𝑎 3 𝑎 5 (𝑁) 𝑎 0 𝑘=1 𝑘 𝑡, 𝑛 𝜔 𝑘=2 𝜔 0 3𝜔 0 5𝜔 0 𝑡, 𝑛 2𝜔 0 4𝜔 0 𝑘=3 All with period T: integer multiples of ω0 Discrete in frequency domain (𝑁)

Continuous/Discrete Sinusoidals
(P.36 of 1.0)

Exponential/Sinusoidal Signals
Harmonically related discrete-time signal sets all with common period N This is different from continuous case. Only N distinct signals in this set. (P.42 of 1.0)

Determination of ak (P.14 of 3.0) Fourier series coefficients dc component

𝐴 ⋅ 𝑣 𝑛 =( 𝑘 𝑎 𝑘 𝑣 𝑘 )⋅ 𝑣 𝑛 𝑣 𝑘 ⋅ 𝑣 𝑛 = 𝑇, 𝑘=𝑛 0,𝑘≠𝑛 Not unit vector
(P.15 of 3.0) 𝐴 ⋅ 𝑣 𝑛 =( 𝑘 𝑎 𝑘 𝑣 𝑘 )⋅ 𝑣 𝑛 𝑣 𝑘 ⋅ 𝑣 𝑛 = 𝑇, 𝑘=𝑛 0,𝑘≠𝑛 Not unit vector orthogonal 𝐴 ⋅ 𝑣 𝑛 =𝑇 𝑎 𝑛 𝑎 𝑛 = 1 𝑇 ( 𝐴 ⋅ 𝑣 𝑛 ) (分析)

repeat with period N Note: both x[n] and ak are discrete, and periodic with period N, therefore summed over a period of N 𝐴 = 𝑘 𝑎 𝑘 𝑣 𝑘 (合成) 𝑎 𝑘 = 𝐴 ⋅ 𝑣 𝑘 (分析) N different values in 𝑥[𝑛] N-dimensional vector space

Orthogonal Basis

Vector Space Interpretation is a vector space

Vector Space Interpretation a set of orthonormal bases

No Convergence Issue, No Gibbs Phenomenon, No Discontinuity x[n] has only N parameters, represented by N coefficients sum of N terms gives the exact value N odd – N even See Fig. 3.18, P.220 of text

Properties Primarily Parallel with those for continuous-time
Multiplication periodic convolution 𝑎 𝑗 𝑏 𝑘−𝑗 𝑗

Time Shift First Difference 𝑥 𝑡− 𝑡 0 ⟷ 𝑒 −𝑗𝑘 𝜔 0 𝑡 0 𝑎 𝑘
𝑥 𝑡− 𝑡 0 ⟷ 𝑒 −𝑗𝑘 𝜔 0 𝑡 𝑎 𝑘 𝑥 𝑛− 𝑛 0 ⟷ 𝑒 −𝑗𝑘 𝜔 0 𝑛 𝑎 𝑘 𝑥 𝑛−1 ⟷ 𝑒 −𝑗𝑘( 2𝜋 𝑁 ) 𝑎 𝑘 First Difference

Properties Parseval’s Relation average power in a period
average power in a period for each harmonic component

System Characterization
3.5 Application Example System Characterization y[n], y(t) h[n], h(t) x[n], x(t) δ[n], δ(t) zn , e st H(z)zn, H(s)est, z=e jω, s=j e jωn , e jωt H(e jω)e jωn, H(j)e jωt  n  n

Superposition Property
Continuous-time Discrete-time H(j), H(ejω) frequency response, or transfer function

Filtering modifying the amplitude/ phase of the different frequency components in a signal, including eliminating some frequency components entirely frequency shaping, frequency selective Example 1 See Fig. 3.34, P.246 of text

Filtering Example 2 See Fig. 3.36, P.248 of text

Examples Example 3.5, p.193 of text

Examples Example 3.5, p.193 of text (a) (b) (c)

Examples Example 3.8, p.208 of text (a) (b) (c)

Examples Example 3.8, p.208 of text

Examples Example 3.17, p.230 of text y[n], y(t) h[n], h(t) x[n], x(t)
δ[n], δ(t) x[n] y[n] h[n]

Problem 3.66, p.275 of text .

Problem 3.70, p.281 of text 2-dimensional signals

3.0 Fourier Series Representation of Periodic Signals

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