Signals and Systems Lecture 5: Discrete Fourier Series Dr. Guillaume Ducard Fall 2018 based on materials from: Prof. Dr. Raffaello D’Andrea Institute for Dynamic Systems and Control ETH Zurich, Switzerland 1 / 27 Outline 1 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Properties of the discrete Fourier series DFS coefficients of real signals 2 Response to Complex Exponential Sequences Complex exponential as input DFS coefficients of inputs and outputs 3 Relation between DFS and the DT Fourier Transform Definition Example 2 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Overview of Frequency Domain Analysis in Lectures 4 - 6 Tools for analysis of signals and systems in frequency domain: The DT Fourier transform (FT): For general, infinitely long and absolutely summable signals. ⇒ Useful for theory and LTI system analysis. The discrete Fourier series (DFS): For infinitely long but periodic signals ⇒ basis for the discrete Fourier transform. The discrete Fourier transform (DFT): For general, finite length signals. ⇒ Used in practice with signals from experiments. Underlying these three concepts is the decomposition of signals into sums of sinusoids (or complex exponentials). 3 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Outline 1 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Properties of the discrete Fourier series DFS coefficients of real signals 2 Response to Complex Exponential Sequences Complex exponential as input DFS coefficients of inputs and outputs 3 Relation between DFS and the DT Fourier Transform Definition Example 4 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Recall on periodic signals A periodic signal displays a pattern that repeats itself, for example over time or space. Recall A periodic sequence x with period N is such that x[n + N]= x[n], ∀n 5 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Overview of the Discrete Fourier Series Analysis equation (Discrete Fourier Series) DFS coefficients are obtained as: N−1 −jk 2π n X[k] = X x[n]e N . n=0 Synthesis equation (Inverse Discrete Fourier Series) N−1 1 jk 2π n x[n]= X[k]e N N X k=0 6 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Discrete Fourier series representation of a periodic signal The Discrete Fourier Series (DFS) is an alternative representation of a periodic sequence x with period N. The periodic sequence x can be represented 2π as a sum of N complex exponentials with frequencies k N , where k = 0, 1,...,N −1: N−1 1 jk 2π n x[n]= X[k]e N (1) N X k=0 for all times n, where X[k] ∈ C is the kth DFS coefficient jk 2π n corresponding to the complex exponential sequence {e N }. 7 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Discrete Fourier series representation of a periodic signal Remarks: 1 2π The frequency N is called the fundamental frequency and is the lowest frequency component in the signal. (We will later show that there is no loss of information in this representation.) 2 We only need N complex exponentials to represent a DT periodic signal with period N. Indeed, there are only N distinct complex exponentials with 2π frequencies that are integer multiples of N : j(k+N) 2π n jk 2π n jN 2π n jk 2π n j2πn jk 2π n e N = e N e N = e N e = e N . 3 graphical representation (shown during class). 8 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Discrete Fourier series coefficients The DFS coefficients of Equation (1) are obtained from the periodic signal x (the role of X and x are permuted with a minus sign in the exponential) as: N−1 −jk 2π n X[k] = X x[n]e N . n=0 The DFS operator is denoted as Fs , where: X = Fsx −1 and x = Fs X. The pairs are usually denoted as {x[n]} ←→ {X[k]}. 9 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Discrete Fourier series coefficients Note that, like the underlying sequence x, X is periodic with period N: Proof: N−1 −j(k+N) 2π n X[k + N]= X x[n]e N n=0 N−1 −jk 2π n −j2πn = X x[n]e N e = X[k]. n=0 |=1{z∀n} Conclusions When working with the DFS, it is therefore common practice to only consider one period of the sequence {X[k]}, that is: only the N DFS coefficients as X[k] for k = 0, 1,...,N −1. 10 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Proof that the DFS operator in invertible - Part I The following identity highlights the orthogonality of complex exponentials: N−1 2 Z 1 j(r−k) π n 1 for r − k = mN, m ∈ e N = N n=0 (0 otherwise. X Proof: Case 1 : r − k = mN. In this case, we have 2 jmN π n j2πmn e N = e = 1 for all m, n N−1 N−1 2 1 j(r−k) π n 1 1 ∴ e N = 1= N= 1. N N N n=0 n=0 X X Case 2 : r − k 6= mN. Define l := r − k. We then have 2 N−1 jl π N 1 2π 1 1 − e N jl N n e = 2π , N N jl N n=0 1 − e X jl 2π the above equation is a geometric series and e N 6= 1 since l 6= mN. For 2 1 1−ej πl l mN 2 . 6= , we therefore have N jl π = 0 1−e N 11 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Proof that the DFS operator in invertible - Part II −1 In order to prove that Fs is the inverse transform of Fs , we need to show: −1 1 Fs Fs = I −1 2 and Fs Fs = I, where I is the identity operator. −1 We now show that Fs Fs = I: N−1 N−1 2 2 −1 1 jr π n −jk π n F F {X[k]} = X[r]e N e N s s N ( n=0 r=0 ! ) X X N−1 N−1 2 1 j(r−k) π n = X[r] e N . N r=0 n=0 X X From above equation, the term in underbrace is equal to 1 forr = k mod N and 0 otherwise. Thus: | {z } N−1 N−1 2 −1 1 j(r−k) π n F F {X[k]} = X[r] e N = {X[k mod N]} = {X[k]}. s s N ( r=0 n=0 !) X X 12 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals −1 In a similar way, it can also be shown that Fs Fs = I. Remark: As both x and X are periodic with period N, we can sum over any N consecutive values (denoted as hNi): 1 jk 2π n −jk 2π n x[n]= X[k]e N and X[k]= x[n]e N . N X X k=hNi n=hNi 13 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Outline 1 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Properties of the discrete Fourier series DFS coefficients of real signals 2 Response to Complex Exponential Sequences Complex exponential as input DFS coefficients of inputs and outputs 3 Relation between DFS and the DT Fourier Transform Definition Example 14 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Properties of the discrete Fourier series Linearity a1{x1[n]} + a2{x2[n]} ←→ a1{X1[k]} + a2{X2[k]} Parseval’s theorem (recall: period is N) N−1 N−1 1 |x[n]|2 = |X[k]|2 X N X n=0 k=0 15 / 27 The Discrete Fourier Series Discrete Fourier series representation of a periodic signal Response to Complex Exponential Sequences Properties of the discrete Fourier series Relation between DFS and the DT Fourier Transform DFS coefficients of real signals Proof of Parseval’s theorem N−1 1 |X[k]|2 =? N k X=0 N−1 1 ∗ ∗ = X [k]X[k] (where denotes the complex conjugate) N k X=0 N−1 N−1 1 ∗ jk 2π n = x [n]e N X[k] N k n=0 ! X=0 X N−1 N−1 2 1 ∗ jk π n = x [n]X[k]e N N k n=0 X=0 X N−1 N−1 ∗ 1 jk 2π n = x [n] X[k]e N