Chaper 2 : Fourier Series.0 Inroducion Fourier Series : represenaion of periodic signals as weighed sums of harmonically relaed frequencies. If a signal x() is periodic signal, hen x() can be represened in erms of Fourier Series eiher in : a) Trigonomeric Form b) Exponenial (Complex) Form Fourier Series Saw ooh waveform Infinie number of sine waves (harmonics) Chaper 3 : Fourier Series. Trigonomeric Fourier Series General equaion : x() is expressed as he sum of sinusoidal componens having differen frequencies where : * a n and b n : he Fourier coefficiens * a 0 : DC Value
Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as Trigonomeric Fourier Series: x() -T -0.5T 0 0.5T T - Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as Trigonomeric Fourier Series: x() x() = / ; 0 < < 0 ; < < 2-2 - 0 2 3-2
Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as Trigonomeric Fourier Series: x() -2-0 2 3 - Chaper 3 : Fourier Series.2 Symmery Properies The symmery properies can be classified ino 5 ypes : a) Even Symmery b) Odd Symmery c) Half Wave Symmery d) Even And Half Wave Symmery e) Odd And Half Wave Symmery 3
Chaper 3 : Fourier Series.2 Symmery Properies a) Even Symmery The signal x() is said o be even symmery if x() = x(-) x() x() = x(-) x() x() = x(-) Chaper 3 : Fourier Series.2 Symmery Properies b) Odd Symmery The signal x() is said o be even symmery if x() = -x(-) x() x() = -x(-) x() - - x() = x(-) 4
Chaper 3 : Fourier Series.2 Symmery Properies c) Half Wave Symmery The signal x() is said o be half wave symmery if x() = -x( + T/2) x() x() = -x(+t/2) - T/2 - x() x() = -x(+t/2) T/2 T T/2 T/2 T Chaper 3 : Fourier Series The firs half-cycle of a periodic signal is shown in figure below and he period is T = 2 sec. Skech y() clearly for 3 complee cycles if : i) y() is an even-symmerical signal ii) y() is an odd-symmerical signal iii) y() is a half-wave symmerical signal x() 5
Chaper 3 : Fourier Series Answer i) even-symmerical signal iii) odd-symmerical signal x(-) -x(-) -3-2 - 0 2 3-3 -2-0 2 3 - Chaper 3 : Fourier Series Answer iii) half-wave symmerical signal x-(+) -3-2 - 0 2 3-6
Chaper 3 : Fourier Series The firs half-cycle of a periodic signal is shown in figure below and he period is T = 4 sec. Skech y() clearly for 3 complee cycles if : i) y() is an even-symmerical signal ii) y() is an odd-symmerical signal iii) y() is a half-wave symmerical signal 3 y() 2 Chaper 3 : Fourier Series Answer i) even-symmerical signal iii) odd-symmerical signal x(-) 3 -x(-) 3-5 -4-3 -2-2 3 4 5-5 -4-3 -2-2 3 4 5 - -3 7
Chaper 3 : Fourier Series Answer iii) half-wave symmerical signal x-(+) 3-5 -4-3 -2-2 3 4 5-3 Chaper 3 : Fourier Series.2 Symmery Properies d) Even And Half-Wave Symmery Consider a half cycle signal x() shown in figure below where T = 4 sec x() - For one complee cycle, he shape of signal x() is he same properies. x(-) -x(+t/2) - - - - 8
Chaper 3 : Fourier Series.2 Symmery Properies d) Even And Half-Wave Symmery Thus, he half-wave even symmery signal x() for 3 complee cycles is shown below : x() = half-wave even -6-5 -4-3 -2-0 - 2 3 4 5 6 Chaper 3 : Fourier Series.2 Symmery Properies e) Odd And Half-Wave Symmery Consider a half cycle signal x() shown in figure below where T = 4 sec x() 2 - For one complee cycle, he shape of signal x() is he same properies. -x(-) -x(+t/2) -2-2 -2-2 - - 9
Chaper 3 : Fourier Series.2 Symmery Properies e) Odd And Half-Wave Symmery Thus, he half-wave even symmery signal x() for 3 complee cycles is shown below : x() = half-wave odd -6-5 -4-3 -2-0 - 2 3 4 5 6 Chaper 3 : Fourier Series.3 Effecs of Symmery Properies n = odd 0
Chaper 3 : Fourier Series.3 Effecs of Symmery Properies Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as Trigonomeric Fourier Series using symmery propery. x() -T -0.5T 0 0.5T T -
Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as Trigonomeric Fourier Series using symmery propery. x() -T -0.5T 0.5T T - Chaper 3 : Fourier Series.4 Hidden Symmery Express signal x() as TFS x() A -2T -T T 2T Soluion x() does no posses any symmery properies. 2
Chaper 3 : Fourier Series.4 Hidden Symmery Soluion However applying he hidden symmery by shifing 0.5A of he DC value for signal x() will resul signal below : g() A -2T -T 0.5A T 2T Now, signal g() posses odd symmery, hus FS of x() : x() = 0.5 A + FS of signal g() Chaper 3 : Fourier Series.4 Hidden Symmery Express signal x() as TFS x() π -3π -2π π 0 π 2π 3π 3
Chaper 3 : Fourier Series.4 Hidden Symmery Soluion Signal x() posses even-symmery propery. Thus FS of x() is : x() = π/2 + FS of signal g() x() π -3π -2π π 0 π 2π 3π Chaper 3 : Fourier Series.5 Trigonomeric Fourier Series The rigonomeric FS of signal x() is given as : Euler s Ideniy The expression ( ) can be expressed as follows : 4
Chaper 3 : Fourier Series.5 Complex Fourier Series Thus x() : Le : Then x() : Chaper 3 : Fourier Series.5 Complex Fourier Series The erm can also be represened as follows : = = Then, x() = + + 5
Chaper 2 : Fourier Series.5 Complex Fourier Series Where : where : (Euler s Ideniy) Chaper 3 : Fourier Series Express he following signal x() as shown in figure below as an Exponenial Fourier Series. x() -T -0.5T 0 0.5T T - 6
Chaper 2 : Fourier Series Express he following signal x() as shown in figure below as an Exponenial Fourier Series. x() -2T -T T 2T 3T Chaper 2 : Fourier Series The following figure below shows a half cycle of a periodic signal x(). If x() is an odd symmery signal, deermine is EFS : x() 2 2 7
Chaper 2 : Fourier Series Soluion x() is an odd symmery signal x() 2 T = 4 ω 0 = 2π / 4 = π / 2 Odd symmery -4-2 2 4 a 0 = 0 (C 0 ) a n = 0-2 b n =???? Chaper 2 : Fourier Series.6 Frequency Specrum Frequency specrum consis of : a) Ampliude specrum b) Phase specrum he plo of І C n І agains nω 0 he plo of C n agains nω 0 І C n І C n nω 0 nω 0 8
Chaper 2 : Fourier Series.6 Frequency Specrum For complex number Im (jω) І C n І a + jb Magniude, І C n І = Phase C n = C n Real (σ) Chaper 2 : Fourier Series.6 Frequency Specrum Condiions i) If b = 0, C n = a ii) If a = 0, C n = jb jω jω jb π/2 σ -π/2 -jb σ -a a І C n І = a І C n І = b C n = 0 if a > 0 (posiive) = π/2 if b > 0 (posiive) = π or π if a < 0 (negaive) = -π/2 if b < 0 (negaive) C n 9
Chaper 2 : Fourier Series Express he following signal x() as shown in figure below as an Exponenial Fourier Series. Plo he frequency specrum for signal x(). x() -T -0.5T 0 0.5T T - Chaper 2 : Fourier Series Express he following signal x() as shown in figure below as an Exponenial Fourier Series. Plo he frequency specrum of signal x(). x() -2T -T T 2T 3T 20
Chaper 2 : Fourier Series The Fourier Series of signal x() is given as : Skech he frequency specrum of x() for n = ±5, ±4, ±3, ±2, ± and 0 Chaper 2 : Fourier Series.7 TFS Coefficiens and EFS Coefficiens Relaionship or 2
Chaper 2 : Fourier Series Conver he TFS coefficiens of signal x() of figure below o EFS coefficiens. x() From previous TFS a 0 = 0 -T -0.5T 0-0.5T T a n = 0 b n = 0 ; n = even 4 / nπ ; n = odd Chaper 2 : Fourier Series Conver he TFS coefficiens of signal x() of figure below o EFS coefficiens. π x() From previous TFS a 0 = π / 2 b n = 0-3π -2π π 0 π 2π 3π a n = 0 ; n = even -4/n 2 π ; n = odd 22
Chaper 2 : Fourier Series Consider he signal y() as shown below : i) Deermine he TFS of y() ii) From he TFS of y(), wrie he EFS of y() x() 2-4 -3-2 - 0 2 3 4 - -2 23