Lecture 06 Lecture 07 Lecture 08 Lecture 09 EE150 Signals and Systems – Part 3: Fourier Series Representation of Periodic Signals 1 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Objective Recall: Previously, we use the weighted sum (integral) of shifted impulses to represent an input and then derive the convolution sum (integral). F In this chapter: We use different basic signal, the complex exponential, to represent the input. F Why we use complex exponential? 2 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Eigenfunction of LTI System A signal for which the system output is just a constant (possibly complex) times the input is referred to as an eigenfunction of the system. Objective The output to an input x(t) can be found easily if x(t) can be expressed as weighted sum of the eigenfunctions. 3 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Eigenfunction of CT LTI Systems Consider an input x(t) = est and a CT LTI system with impulse response h(t) Z ∞ y (t) = h(τ )es(t−τ ) dτ −∞ Z ∞ = est h(τ )e−sτ dτ −∞ R∞ Let H(s) = −∞ h(τ )e−sτ dτ (eigenvalue), then y (t) = H(s)est . Hence, complex exponentials are eigenfunctions of LTI systems: est → H(s)est 4 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Eigenfunction of DT LTI Systems Consider an input x[n] = z n and a DT LTI system with impulse response h[n] y [n] = ∞ X h[k]z n−k =z n k=−∞ ∞ X h[k]z −k k=−∞ P −k (eigenvalue), then y [n] = H(z)z n . Let H(z) = ∞ k=−∞ h[k]z Hence, complex exponentials are eigenfunctions of LTI systems: z n → H(z)z n 5 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Usefulness of Complex Exponentials Ex: If input x(t) = a1 es1 t + a2 es2 t + a3 es3 t , based on eigenfunction property and superposition property, the the response is y (t) = a1 H(s1 )es1 t + a2 H(s2 )es2 t + a3 H(s3 )es3 t Generally, for a CT LTI system, if the input is a linear combination of complex exponentials, then X X x(t) = ak esk t → y (t) = ak H(sk )esk t . k k Similarly, for a DT LTI system, X X x[n] = ak zkn → y [n] = ak H(zk )zkn . k k 6 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Analysis For Fourier analysis, we consider 1 CT: purely imaginary s = jω: ejωt 2 DT: unit magnitude z = ejω : ejωn ejωn → H(ejω )ejωn 7 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Periodic signals & Fourier Series Expansion Using “trigonometric sum” to describe periodic signal can be tracked back to Babylonians who predicted astronomical events similarly. L. Euler (in 1748) and Bernoulli (in 1753) used the “normal mode concept to describe the motion of a vibrating string; though JL Lagrange strongly criticized this concept. Fourier (in 1807) had found series of harmonically related sinusoids to be useful to describe the temperature distribution through body, and he claimed any periodic signal can be represented by such series. Dirichlet (in 1829) provide a precise condition under which a periodic signal can be represented by a Fourier series. 8 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Aside: an orthonormal set Consider the set, S, of x(t) satisfying x(t) = x(t + T0 ) Dot-product (inner-product) defined as ω0 < x1 (t), x2 (t) >= 2π Z π ω0 − ωπ 0 x1 (t)x2∗ (t)dt Consider the set, B, of functions in S φk (t) = e jkω0 t ; ω0 = 2π ,k ∈ Z T0 Observe that they are orthonormal ω0 2π Z π ω0 − ωπ 0 e jk1 ω0 t e −jk2 ω0 t dt = ( 0 k1 6= k2 1 k1 = k2 9 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier’s Idea The span of the orthonormal functions, B, covers most of S. i.e. span(B) ≈ S More precisely, under mild assumptions: x(t) is sum of sinusoids, i.e. Z π X ω0 ω0 x(τ )e −jkω0 τ dτ x(t) = ak e jkω0 t , where ak = 2π − π k ω0 10 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example P Consider a periodic signal x(t) = 3k=−3 ak ejk2πt , where a0 = 1, a1 = a−1 = 0.25, a2 = a−2 = 0.5, a3 = a−3 = 1/3. Collecting each of the harmonic components having the same fundamental frequency x(t) = 1+ 1 1 1 j2πt e + e−j2πt + ej4πt + e−j4πt + ej6πt + e−j6πt 4 2 3 Using Euler’s relation x(t) = 1 + 1 2 cos(2πt) + cos(4πt) + cos(6πt) 2 3 11 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example cont. 12 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Periodic signals & Fourier Series Expansion cont. Theorem (for reasonable functions): x(t) may be expressed as a Fourier series: x(t) = ∞ X ak · e jkω0 t k=−∞ (sum of sinusoids whose frequencies are multiple of ω0 , the “fundamental frequency”.) Where ak can be obtained by Z 1 ak = x(τ )e −jkω0 τ dτ - Fourier series coefficient. T 0 T0 Note: e jkω0 t , for k = −∞ to ∞, are orthonormal function. (Normal basic signal) 13 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.4 Let x(t) = 1 + sin(ω0 t) + 2 cos(ω0 t) + cos(2ω0 t + π4 ) has fundamental frequency ω0 . First, expanding x(t) in terms of complex exponentials π 1 1 π 1 1 ejω0 t + 1 − e−jω0 t + ej 4 ej2ω0 t + e−j 4 e−j2ω0 t x(t) = 1+ 1 + 2j 2j 2 2 Then, the Fourier series coefficients can be directly obtained. 14 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.5 Square Wave For a periodic square wave, the definition over one period is 1, |t| < T1 , x(t) = 0, T1 < |t| < T2 . Periodic with fundamental period T and fundamental frequency ω0 = 2π T 15 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.5 Square Wave cont. For k = 0, 1 a0 = T Z T1 dt = −T1 2T1 . T For k 6= 0, 1 ak = T Z T1 e−jkω0 t dt = −T1 2 sin(kω0 T1 ) sin(kω0 T1 ) = . kω0 T kπ When T1 = 41 T → 50% duty-cycle square wave ak = sin( kπ 2 ) . kπ 16 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.5 Square Wave cont. 17 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series x(t) = = ∞ X ak e jkω0 t k=−∞ ∞ X (ak cos(kω0 t) + jak sin(kω0 t)) k=−∞ = a0 + X ((ak + a−k )cos(kω0 t) + j(ak − a−k )sin(kω0 t)) k>0 In general, ak is complex. Therefore, this is not the real and imaginary decomposition. However, this is the even and odd decomposition 18 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series cont. 1 Odd/Even periodic functions 19 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series cont. 2 Approximation by Truncating Higher Harmonics If ak (for |k| > N) are small, x(t) ≈ x̂(t) ≡ N X ak e jkω0 t −N The approximation error is e(t) = x(t) − x̂(t) How good is the approximation? Metric: relative energy in the error over a period R < e(t), e(t) > err = =R < x(t), x(t) > π ω0 −π ω0 π ω0 −π ω0 e(t)e ∗ (t)dt x(t)x ∗ (t)dt |k|>N |ak |2 k=−∞ |ak |2 P = P∞ 20 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Convergence of Fourier Series Conditions for convergence of Fourier series is a deep subject - Finite energy over a single period - Dirichlet conditions: Over any period, x(t) must be absolutely integrable Finite number of extrema during any single period Finite number of discontinuities in any finite interval of time; each of these discontinuities is finite. Check: https://en.wikipedia.org/wiki/Dini_test 21 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Signals Violating Dirichlet Conditions 22 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Gibbs Phenomenon Overshoot ≈ 9% as N goes to ∞ 23 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Properties of Continuous-Time Fourier Series Assume x(t) is periodic with period T and fundamental frequency ω0 = 2π/T . x(t) and its Fourier-series coefficients ak are denoted by x(t)←−FS −−→ak Assume x(t)←−FS −−→ak , y (t)←−FS −−→bk (using same T ) 1 Linearity: z(t) = αx(t) + βy (t)←−FS −−→αak + βbk 2 Time-shift: −jkω0 t0 a x(t − t0 )←−FS k −−→e 24 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Properties of C-T Fourier Series cont. 3 Time-reverse: x(−t)←−FS −−→a−k 4 Time-scaling: ∞ P x(αt) = ak e jk(αω0 )t k=−∞ 5 Multiplication: x(t)y (t)←−FS −−→hk = ∞ P al bk−l − Convolution! l=−∞ 25 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Properties of C-T Fourier Series cont. 6 conjugation & conjugate symmetry: ∗ x ∗ ←−FS −−→a−k → x(t) = x ∗ (t) → ak∗ = a−k If x(t) is real and even, → ak = a−k = ak∗ → Fourier Series Coefficients are real & even If x(t) is real and odd, → ak = −a−k = −ak∗ → Fourier Series Coefficients are purely imaginary & odd If x(t) real 26 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Properties of C-T Fourier Series cont. 7 dx(t) FS dt ←− −−→jkω0 ak 8 Parseval’s Identity: Rt ak FS −∞ x(τ )dτ ←− −−→ jkω0 1 T R T |x(t)|2 dt = ∞ P |ak |2 k=−∞ Proof. 1 T Z T Z X 1 ak1 ak∗2 e j(k1 −k2 )ω0 t dt T T k1 ,k2 X = ak1 ak∗2 δ[k1 − k2 ] |x(t)|2 dt = k1 ,k2 = X |ak |2 k 27 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example: a x(t) = cos ω0 t → x(t) = 1 2 e jω0 t + e −jω0 t = 12 e jω0 t + 12 e −jω0 t ∴ a0 = 0, a1 = 21 , a−1 = 12 , ak = 0 otherwise b 28 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example: cont. Recall from the previous example, we have By changing of variable, we can obtain 29 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example: cont. Assume: a(t)←−FS −−→ak , ⇒ ck = ak e b(t)←−FS −−→bk , 1 a0 − , 2 T −jkω0 40 ∴ ck = , k=0 where ω0 = 2π k 6= 0 0 π sin(k 2 ) kπ c(t)←−FS −−→ck × e −j kπ 2 , k=0 , k 6= 0 1 T0 30 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example: cont. 0 Assume: g (t)←−FS −−→dk 0 ∵ g (t) = 2 · c(t) with T0 = 2 ∴ dk = 2 · ck Assume: g (t)←−FS −−→ek ∴ ek = For k 6= 0 ⇒ ek = For k = 0 e0 = dk 2ck 2ck = = jkω0 jkω0 jkπ T20 2 sin( kπ ) −j kπ 2 e 2 j(kπ)2 (T0 = 2) The area under g (t) in one period period = 1 2 31 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.9 Suppose we have the following facts about a signal x(t) 1 x(t) is a real signal 2 x(t) is periodic with period T = 4, and it has Fourier series coefficient ak 3 ak = 0 for k > 1 4 5 The signal with Fourier coefficient ck = e−jπk/2 a−k is odd R 1 1 2 4 4 |x(t)| dt = 2 32 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series for Discrete-Time Periodic Signal Difference: 1 2 continuous-time: infinite series discrete-time: finite series no convergence issue in discrete-time F Recall: x[n] is periodic with period N if x[n] = x[n + N] The fundamental period is the smallest positive N which satisfies the above eq.; and ω0 = 2π/N is the fundamental frequency. 33 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 An Orthonormal Set Consider the set, T , of x[n] satisfying x[n] = x[n + N] Dot-product (inner-product) defined as N−1 1 X < x1 [n], x2 [n] >= x1 [n]x2∗ [n] N n=0 Consider the set, C , of N functions in T µk [n] = e jkω0 n ; ω0 = 2π ,0 ≤ k ≤ N − 1 N Observe that they are orthonormal N−1 1 X jk1 ω0 m −jk2 ω0 m e e = N m=0 ( 0 k1 6= k2 1 k1 = k2 34 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series for Discrete-Time Periodic Signal Theorem x[n] = N−1 X ak e jkω0 n , where ak = N−1 1 X x[m]e −jkω0 m N m=0 k=0 Proof: x[n] = ⇔ ⇔ N−1 X N−1 1 X x[m]e −jkω0 m e jkω0 n , N k=0 N−1 X 1 x[n] = N x[n] = m=0 N−1 X m=0 x[m] N−1 X e −jkω0 m e jkω0 n , k=0 x[m]δ[n − m]. m=0 35 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.12 Discrete-time periodic square wave ak = = N1 1 X e−jk(2π/N)n N n=−N1 ( 1 sin[2πk(N1 +1/2)/N] ,k N sin(πk/N) 2N1 +1 k N , 6= 0, ±N, ±2N, . . . = 0, ±N, ±2N, . . . 36 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Properties of Discrete-Time Fourier Series x[n] and y [n] are periodic with period N, ak and bk are periodic with period N FS 1 Linearity: Ax[n] + By [n] ←→ Aak + Bbk 2 Time shifting: x[n − n0 ] ←→ ak e−jk(2π/N)n0 3 Frequency shifting: ejM(2π/N)n x[n] ←→ ak−M 4 ∗ Conjugation: x ∗ [n] ←→ a−k 5 6 7 8 9 FS FS FS FS Time reversal: x[−n] ←→ a−k P FS Periodic convolution: r =<N> x[r ]y [n − r ] ←→ Nak bk FS P Multiplication: x[n]y [n] ←→ l=<N> al bk−l FS First difference: x[n] − x[n − 1] ←→ (1 − e−jk(2π/N) )ak P P Parseval’s relation: N1 n=<N> |x[n]|2 = k=<N> |ak |2 37 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Example 3.13 As x[n] = x1 [n] + x2 [n], we have ak = bk + ck 38 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series and LTI-System Recall: eigenfunction For discrete-time, similarly we have 39 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series and LTI-System cont. H(s) and H(z) are referred to as “system function” (transfer functions). (C − T ) continuous-time: Re{s} = 0 → s = jω (D − T ) discrete-time: |z| = 1 → z = e jω R∞ H(jω) = −∞ h(t)e −jωt dt, C − T , ∞ P ⇒ H(e jω ) = h[n]e −jωn , D − T . n=−∞ H(jω) and H(e jω ) are the “frequency response” of the continuous time and discrete-time system, respectively. 40 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Remarks The transfer function, H(jω), is then H(jω) ≡ |H(jω)| e jφ with φ ≡ ∠H(jω) The output is → H(jω)e jωt = |H(jω)| e j(ωt+φ) 41 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Fourier Series and LTI-System Note: For C.T., periodic signal, then ∞ X x(t) = k=−∞ ak e jkω0 t → y (t) = ∞ X ak H(jkω0 )e jkω0 t k=−∞ Similarly, X X x[n] = ak ejk(2π/N)n → y [n] = ak H(e j2πk/N )e jk(2π/N)n k=<N> k=<N> 42 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Remarks cont. 1 Sinusoids are eigenfunctions for any LTI system. 2 H(jω) characterizes the “frequency response” of a linear system. Z ∞ H(jω) ≡ h(τ )e −jωτ dτ −∞ H(jω) is said to be the Fourier Transform of the time domain function h(t). Both h(t) and H(jω) can be used to find the output for a particular input. 43 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Filter Filtering: A process that changes the relative amplitude (or phase) of some frequency components. F e.g. frequency-shaping filter (like equalizer in a Hi-Fi system) frequency-selective filter (like low-pass, band-pass, high-pass filters) 44 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Frequency Shaping Filter E.g. Differentiator (a HPF or LPF?) y (t) = dx(t) → H(jω) = jω dt →high frequency component is amplified & low frequency component is suppressed.→ HPF (high pass filter) (a.k.a. as edge-enhancement filter in image processing) 45 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Frequency-Selective Filter Select some bands of frequencies and reject others. 1 46 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Frequency-Selective Filter cont. 2 D-T 47 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Frequency-Selective Filter cont. For D-T: H(e jω ) is periodic with period 2π low frequencies: at around ω = 0, ±2π, ±4π, . . . high frequencies: at around ω ± π, ±3π, . . . Note: ideal filters are not realizable; Practical filters have transition band, and may have ripple in stopband and passband 48 / 49 Lecture 06 Lecture 07 Lecture 08 Lecture 09 Summary Developed Fourier series representation for both C-T and D-T systems. Properties of Fourier Series Eigenfunction and Eigenvalue of LTI systems System function (transfer function) and frequency response Filtering 49 / 49