Overview of DT Fourier Series Topics Orthogonality of DT exponential - PowerPoint PPT Presentation

Overview of DT Fourier Series Topics • Orthogonality of DT exponential harmonics • DT Fourier Series as a Design Task • Picking the frequencies • Picking the range • Finding the coefficients • Example J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 1

Motivation h [ n ] H (e j Ω ) x [ n ] y [ n ] x [ n ] y [ n ] e j Ω n → H (e j Ω n )e j Ω n X [ k ] e j Ω k n → � � X [ k ] H (e j Ω k n )e j Ω k n k k ∞ � H (e j Ω ) = F { h [ n ] } = h [ n ] e − j Ω n n = −∞ • For now, we restrict out attention to DT periodic signals: x [ n + N ] = x [ n ] • Would like to represent x [ n ] as a sum of complex sinusoids • Why? Gives us insight and simplifies computation J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 2

Example 1: Complex Sinusoidal Sum To solve for the DTFS coefficients, we need the following relation � k = ℓN, N e jk Ω n = � 0 k � = ℓN n = <N> for any integer ℓ . Prove that this relation is true. Hint: recall the finite sum formula N − 1 a n = 1 − a N � 1 − a n =0 J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 3

Example 1: Workspace J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 4

DT Periodic Signals Design Task x [ n ] 1 n -4 -3 -2 -1 0 1 2 3 4 � 1 . 0 n = 3 k x [ n ] = 0 . 5 Otherwise • Suppose we have a DT signal x [ n ] that we know is periodic • The signal is applied at the input of an LTI system • We would like to estimate the signal as a sum of complex sinusoids � X [ k ] e j Ω k n x [ n ] = ˆ k J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 5

DT Periodic Signals Design Task � X [ k ] e j Ω k n x [ n ] = ˆ k • Here the ˆ symbol is used to indicate that the sum is an approximation (estimate) of x [ n ] • Enables us to calculate the system output easily • Must pick – The frequencies Ω k – The range of the sum � k – The coefficients X [ k ] J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 6

Design Task: Picking the Frequencies � X [ k ] e j Ω k n x [ n ] = ˆ k • We know x [ n ] is periodic with some fundamental period N • If ˆ x [ n ] is to approximate x [ n ] accurately, it should also repeat every N samples • In order for ˆ x [ n ] to be periodic with period N , every complex sinusoid must also be periodic • Only a harmonic set of complex sinusoids have this property • Thus Ω k = k Ω where Ω = 2 π N � X [ k ] e jk Ω n x [ n ] = ˆ k J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 7

Design Task: Picking the Range � X [ k ] e jk Ω n x [ n ] = ˆ k • Recall that there are only N distinct complex sinusoids that repeat every N samples e jk Ω n = e j ( k + ℓN ) n where Ω = 2 π N • Thus we can pick the range of the sum so that it includes only these terms • Typical choices ( N − 1) / 2 N − 1 � � X [ k ] e jk Ω n X [ k ] e jk Ω n x [ n ] = ˆ x [ n ] = ˆ k =0 k = − ( N − 1) / 2 J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 8

Equivalent Expressions for Exponential Sums � X [ k ] e jk Ω n x [ n ] = k Since only N of the harmonics are distinct, we may truncate the sum so that it only contains only N distinct terms. It does not matter which set of N terms are used. N − 1 N − 1+ ℓ X [ k ] e jkωn = X [ k ] e jkωn = � � � X [ k ] e jkωn x [ n ] = k =0 k = ℓ k = <N> Note the new notation: N + ℓ X [ k ] e jkωn � X [ k ] e jkωn for any ℓ � � k = <N> k =1+ ℓ J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 9

Design Task: Picking the Coefficients MSE = 1 x [ n ] | 2 � � X [ k ] e jk Ω n x [ n ] = ˆ | x [ n ] − ˆ N k = <N> n = <N> • We would like to pick the coefficients X [ k ] so that ˆ x [ n ] is as close to x [ n ] as possible • But what is close? • One measure of the difference between two signals is the mean squared error ( MSE ) • There are other measures, but this is a convenient one because we can differentiate it • Since the signal is periodic, the MSE is calculated over a single fundamental period of N consecutive samples • How do we pick the coefficients X [ k ] to minimize the MSE ? J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 10

Design Task: Coefficient Optimization MSE = 1 x [ n ] | 2 � � X [ k ] e jk Ω n x [ n ] = ˆ | x [ n ] − ˆ N n = <N> k = <N> • Solving for the optimal coefficients is difficult • However, suppose that an optimal solution exists such that MSE = 0 • If such a solution exists, we know the sum of errors will also be zero � 0 = x [ n ] − ˆ x [ n ] n = <N> J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 11

Design Task: Solve for the Coefficients x [ n ]e − jℓ Ω n − � � x [ n ]e − jℓ Ω n 0 = ˆ n = <N> n = <N> x [ n ]e − jℓ Ω n = � � x [ n ]e − jℓ Ω n ˆ n = <N> n = <N> � � � x [ n ]e − jℓ Ω n = � � X [ k ] e jk Ω n e − jℓ Ω n n = <N> n = <N> k = <N> � � e jk Ω n e − jℓ Ω n = X [ k ] n = <N> k = <N> � � e j ( k − ℓ )Ω n = X [ k ] k = <N> n = <N> � = X [ k ] ( Nδ [ k − ℓ ]) k = <N> = X [ ℓ ] N J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 12

Design Task: Optimal Coefficients Thus, the coefficient of the ℓ th complex sinusoid that minimize the MSE is X [ ℓ ] = 1 � x [ n ]e − jℓ Ω n N n = <N> • With more algebra, you should be able to show that these coefficients result in MSE = 0 ! • Thus, ˆ x [ n ] = x [ n ] • This means any DT periodic signal can be exactly represented as a sum of N complex sinusoidal harmonics J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 13

DTFS Observations X [ k ] = 1 � � X [ k ] e jk Ω n x [ n ]e − jk Ω n x [ n ] = N k = <N> n = <N> • The first equation is called the synthesis equation • The second equation is called the analysis equation • The coefficients X [ k ] are called the spectral coefficients or Fourier series coefficients of x [ n ] • We denote the relationship of x [ n ] and X [ k ] by FS x [ n ] ⇐ ⇒ X [ k ] • Both are complete representations of the signal: if we know one, we can compute the other • X [ k ] is a function of frequency ( k Ω ) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 14

Further DTFS Observations & Comments X [ k ] = 1 � � X [ k ] e jk Ω n x [ n ]e − jk Ω n x [ n ] = N n = <N> k = <N> • The DTFS transform is special because both sums are finite – This permits us to calculate the DTFS exactly with computers (e.g. MATLAB) and microprocessors – Later we will see how the DTFS is used to compute the other 3 transforms • The Fourier series representation of signals is useful for LTI system design and analysis because we know how LTI systems affect sinusoidal signals J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 15

Example 2: Pulse x [ n ] 1 n -N -m 0 m N What type of symmetry does the signal have? Find the DT Fourier series coefficients. Plot the coefficient spectrum, partial sums of the Fourier series components, and the MSE versus number of coefficients. J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 16

Example 2: Workspace J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 17

Example 2: Coefficient Spectrum Discrete−Time Fourier Series Coefficients X[k] 0.5 0.4 0.3 0.2 X[k] 0.1 0 −0.1 −0.2 −25 −20 −15 −10 −5 0 5 10 15 20 25 kth (harmonic) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 18

Example 2: Partial Fourier Series Partial Fourier Series Approximation (k=0) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −15 −10 −5 0 5 10 15 Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 19

Example 2: Partial Fourier Series Partial Fourier Series Approximation (k=−1 to 1) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −15 −10 −5 0 5 10 15 Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 20

Example 2: Partial Fourier Series Partial Fourier Series Approximation (k=−2 to 2) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −15 −10 −5 0 5 10 15 Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 21

Example 2: Partial Fourier Series Partial Fourier Series Approximation (k=−3 to 3) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −15 −10 −5 0 5 10 15 Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 22

Example 2: Partial Fourier Series Partial Fourier Series Approximation (k=−4 to 4) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −15 −10 −5 0 5 10 15 Time (samples) J. McNames Portland State University ECE 223 DT Fourier Series Ver. 1.04 23