King Fahd University of Petroleum & Minerals Computer Engineering Dept

Transcription

1 King Fahd University of Petroleum & Minerals Computer Engineering Dept COE 4 Data and Computer Communications erm 5 Dr. shraf S. Hasan Mahmoud Rm -4 Ext ashraf@kfupm.edu.sa /7/6 Dr. shraf S. Hasan Mahmoud Lecture Contents. Fourier nalysis a. Fourier Series Expansion b. Fourier ransform c. Ideal Low/band/high pass filters. Introduction to Z-ransform /7/6 Dr. shraf S. Hasan Mahmoud

2 Signals signal is a function representing information Voice signal microphone output Video signal camera output Etc. ypes of Signals nalog continuous-value continuous-time Discrete discrete-value continuous-time Digital predetermined discrete levels much easier to reproduce at receiver with no errors Binary only two predetermined levels: e.g. and /7/6 Dr. shraf S. Hasan Mahmoud 3 Example of Continuous-Value Continuous-time signal s (t) and s (t) are two example of analog signals s (t) s (t) /7/6 Dr. shraf S. Hasan Mahmoud 4 t t

3 Example of Discrete-Value Continuous-time signal d (t) and d (t) are two example of discrete signals d (t) takes more than two levels d (t) takes only two levels - binary d (t) d (t) /7/6 Dr. shraf S. Hasan Mahmoud 5 t Note d (t) extends to ± t ime Domain Representation ime domain representation we plot value (voltage, current, electric field intensity, etc.) versus time Can infer rate of change (speed or frequency) information e.g. s (t) seems faster than s (t) Using calculus terms: rate of change for s (t) > rate of change for s () s (t) d (t) pplies to BOH analog and digital signals t t s (t) d (t) t t /7/6 Dr. shraf S. Hasan Mahmoud 6 3

4 Frequency - Bandwidth s (t) faster than s (t) s (t) contains higher frequencies than those contained in s (t) s (t) and s (t) contain more than one frequency Minimum frequency f min Maximum frequency f max Bandwidth Range of frequencies contained in signal f max f min pplies to BOH analog and digital signals /7/6 Dr. shraf S. Hasan Mahmoud 7 Frequency Bandwidth () For our example signals, assume: S(t): fmin Hz, fmax 5 Hz S(t): fmin 5 Hz, fmax Hz his means: BW for s (t) 5 49 Hz BW for s (t) Hz pplies to BOH analog and digital signals Note that: because s (t) is faster than s (t) it should contain frequencies higher than those in s (t) E.g. s (t) contains frequencies (5,] which do not exist in s (t) /7/6 Dr. shraf S. Hasan Mahmoud 8 4

5 Frequency Bandwidth (3) Consider the discrete signals d (t) and d (t) he function plots have points of infinite slope rate of change frequency herefore for signals that look like d (t) and d (t), fmax Furthermore, BW Example: d (t) contains frequencies from some minimum fmin Hz to fmax Hz pplies to BOH analog and digital signals /7/6 Dr. shraf S. Hasan Mahmoud 9 Example of Signal BW Consider the human speech ypically fmin ~ Hz fmax ~ 35 Hz BW of the human speech signal 3 Hz /7/6 Dr. shraf S. Hasan Mahmoud 5

6 Bandwidth for Systems For a system to respond (amplify, process, x, Rx, etc.) for a particular signal with all its details, the system should have an equal or greater bandwidth compared to that of the signal Example: he system required to process s (t) should have a greater bandwidth than the system required to process s (t) /7/6 Dr. shraf S. Hasan Mahmoud Bandwidth for Systems () Example : consider the human ear system Responds to a range of frequencies only fmin Hz fmax, Hz BW 9,98 Hz It does not respond to sounds with frequencies outside this range Example 3: consider the copper wire It passes (electric) signals only between a certain fmin and a certain fmax he higher the quality of the wire the wider the BW More on Systems BW later! /7/6 Dr. shraf S. Hasan Mahmoud 6

7 Frequency Representation How to represent signals and indicate their frequency content? he X-axis: frequency (in Hertz or Hz) What is the Y-axis then? the answer will be postponed! f max 5 Hz pplies to BOH analog and digital signals f min Hz BW for s (t): range of frequencies f f min 5 Hz BW for s (t): range of frequencies f max Hz f /7/6 Dr. shraf S. Hasan Mahmoud 3 Periodic Signals pplies to BOH analog and digital signals periodic signal repeats itself every seconds Period seconds In calculus terms: s(t) is periodic if s(t) s(t+) for any - < t < For previous examples: s (t), s (t), and d (t) are not periodic however, d (t) is periodic /7/6 Dr. shraf S. Hasan Mahmoud 4 7

8 Periodic Signals () pplies to BOH analog and digital signals periodic signal has a FUNDMENL FREQUENCY f f / where is the period periodic signal may also has frequencies other than the fundamental frequency f fmin < f < fmax x x x x x x BW for periodic s(t) f min f max /7/6 Dr. shraf S. Hasan Mahmoud 5 f Periodic Signals (3) pplies to BOH analog and digital signals Examples of other periodic signals: s 3 (t) t s 5 (t) ~ t s 4 (t) t t /7/6 Dr. shraf S. Hasan Mahmoud 6 8

9 Energy/Power of Signals Energy for any signal is defined as E s s( t) dt pplies to BOH analog and digital signals where the integral is carried over LL range of t In other words, Es is the area under the absolute squared of the signal he unit of energy is Joules /7/6 Dr. shraf S. Hasan Mahmoud 7 Energy/Power of Signals () pplies to BOH analog and digital signals Note that for periodic signal E s is equal to infinity since it is defined on (-, ) However power is FINIE for these type of signals Power is defined as the average of the absolute squared of the signal, i.e. P s he unit of power is Joules/sec or Watt s( t) /7/6 Dr. shraf S. Hasan Mahmoud 8 dt Note the integral can be performed on [,], [-/, /], or any continuous interval of length 9

10 VERY SPECIL nalog Signal function of the form s(t) cos(πft + θ) s(t) cos(θ) - Period t /7/6 Dr. shraf S. Hasan Mahmoud 9 Characteristics of COSINE Completely specified by: mplitude Phase - θ Frequency f s(t ) cos(θ) Periodic signal repeats itself every seconds / f ime to review your trigonometry!! E.g. sin(x) cos(x-π/) /7/6 Dr. shraf S. Hasan Mahmoud

11 Characteristics of COSINE () Energy for this signal, E s infinity Power for this signal, P g / Note P g is dependent only on the amplitude Exercise: Verify the above results using the power formula It contains ONLY ONE frequency f he purest form of analog signals Frequency representation: x BW for s (t) f min f max f Hz /7/6 Dr. shraf S. Hasan Mahmoud f Characteristics of COSINE (3) Very Useful Properties (f /) cos( πft + θ ) dt cos( πnft + θ ) dt cos (πft + θ ) dt / cos (πnft + θ ) dt / cos(πnft)cos(πmft) dt / n m n m /7/6 Dr. shraf S. Hasan Mahmoud

12 Example of Cosine Functions Y (t) has a frequency f of Hz ( ½ sec) n amplitude of 3 P Y 3 / 4.5 Watts Y (t) - has a frequency f of Hz ( / sec) n amplitude of P Y /.5 Watts value Examples of analog contineous-time functions y 3 cos(*pi**t) y cos(*pi**t) time - t /7/6 Dr. shraf S. Hasan Mahmoud 3 ONLY FOR PERIODIC SIGNLS Fourier Series Expansion Can we use the basic cosine functions to represent periodic signals? YES Fourier Series Expansion t s 3 (t) t factorization t t /7/6 Dr. shraf S. Hasan Mahmoud 4

13 Fourier Series Expansion () For a periodic signal s(t) can be represented as a sum of sinusoidal signals as in s( t) + n cos(πnft) + Bn sin(πnf [ t ] ) n where the coefficients are computed using: s( t) dt n s( t)cos(πnf t) dt /7/6 Dr. shraf S. Hasan Mahmoud 5 B f is the fundamental frequency of s(t) and is equal to / n s( t)sin(πnf t) dt Fourier Series Expansion (3) nother form for the series: C s( t) + Cn cos(πnft + θn ) n where the coefficients are computed using: C n n C + B n B n B θ n n θ n tan n n /7/6 Dr. shraf S. Hasan Mahmoud 6 C n 3

14 Notes on Fourier Series Expansion he representation (the sum of sinusoids) is completely identical and equivalent to the original specification of s(t) It is applies to any periodic signal analog or digital! Very powerful tool it reveals all frequencies contained in the original periodic signal s(t) /7/6 Dr. shraf S. Hasan Mahmoud 7 Notes on Fourier Series Expansion () In general, s(t) contains DC term the zero frequency term / (possibly infinite) number of harmonics (or sinusoids) at multiples of the fundamental frequency, f he contribution of a harmonic with frequency nf is proportional to n +B n or C n E.g. if C n ~, then we say the harmonic at nf (or higher does not contribute significantly towards building s(t) more on this when we discuss total power! /7/6 Dr. shraf S. Hasan Mahmoud 8 4

15 Notes on Fourier Series Expansion (3) harmonic with frequency equal to nf (n>), has a period of /(n) In general the series expansion of s(t) contains INFINIE number of terms (harmonics) However for less than % accurate representation one can ignore higher terms terms with frequencies greater than certain n * f /7/6 Dr. shraf S. Hasan Mahmoud 9 Notes on Fourier Series Expansion (4) Lets define the following function: s_e(nk) o be the series expansion of s(t) up to and including the n k term It should be noted that s_e(nk) is periodic with period Examples: s _ e( n ) / s _ e( n ) / + cos(π ft) + B sin(πf t) + C cos(π f t + ) / θ /7/6 Dr. shraf S. Hasan Mahmoud 3 5

16 Notes on Fourier Series Expansion (5) Examples cont d: s _ e( n ) / + cos(πf t) + B sin(πf t) + cos(π f t) + B sin(π f t) / + C cos(π ft + θ) + C cos(π ft + θ) s _ e( n ) [ cos(πnf t) + B sin(πnf t ] + ) n n n / + Cn cos(πnft + θn) n /7/6 Dr. shraf S. Hasan Mahmoud 3 Notes on Fourier Series Expansion (6) It is obvious that s(t) is % represented by s_e(n ) s_e(n n* < ) produces a less than % accurate representation of the original s(t) For most practical periodic signals s_e(n) provides a more than enough accuracy in representing s(t) No need for infinite number of terms /7/6 Dr. shraf S. Hasan Mahmoud 3 6

17 Example : Consider the following s(t) Over one period, the signal is defined as s(t) s(t) -/4 < t < /4 /4 < t < 3/4 -/4 /4 3/4 5/4 t Finding the Series Expansion: he DC term s t dt ( ) /7/6 Dr. shraf S. Hasan Mahmoud 33 Example : cont d he term n : n s( t)cos(πnf t) dt cos(πnft) dt t sin(πnft) πnf t nπ sin( ) πn πn πn n,4,6,... n,5,9,... n 3,7,,... Remember. f /. cos(ax) dx -/a sin(ax) 3. sin(nπ) for integer n 4. sin(nπ/) for n,5,9, 5. sin(nπ/) - for n3,7,, /7/6 Dr. shraf S. Hasan Mahmoud 34 7

18 Example : cont d herefore n is given by: ( ) ( n ) / πn n,4,6,... n,3,5,7,... Remember (-) (n-)/ for n,5,9, - for n 3,7,, /7/6 Dr. shraf S. Hasan Mahmoud 35 Example : cont d he term B n : B n s( t)sin(πnf t) dt sin(πnft) dt t nπ nπ cos(πnft) cos( ) cos( ) πnf t πn Remember. cos(ax)dx -/a sin(ax). cos(x) cos(-x) /7/6 Dr. shraf S. Hasan Mahmoud 36 8

19 Example : cont d herefore, the overall series expansion is given by ( n ) / ( ) s( t) + cos(πnft) π n n,3,5 s( t) + cos(πf t) cos(π 3 ft) π 3π + cos(π 5 ft) cos(π 7 ft) π 7π /7/6 Dr. shraf S. Hasan Mahmoud 37 Example : cont d Original s(t) and the series up to and including n i.e. Comparing: vs. s(t) s_e(n) / original s(t) up to n /7/6 Dr. shraf S. Hasan Mahmoud 38 9

20 Example : cont d Original s(t) and the series up to and including n i.e. Comparing:..8 original s(t) up to n vs. s(t).6.4. s_e(n) / + -. /π cos(πf t) /7/6 Dr. shraf S. Hasan Mahmoud 39 Example : cont d Original s(t) and the series up to and including n 3 i.e. Comparing:..8 original s(t) up to n 3 vs. s(t).6.4. s_e(n3) / + /πcos(πf t) /(3π)cos(π3f t) /7/6 Dr. shraf S. Hasan Mahmoud 4

21 Example : cont d Original s(t) and the series up to and including n i.e. Comparing: vs. s(t) original s(t) up to n s_e(n) / + /πcos(πf t) /(3π)cos(π3f t) + /(5π)cos(π5f t) /(7π)cos(π7f t) +/(π)cos(πf t) /7/6 Dr. shraf S. Hasan Mahmoud 4 Example: cont d clear all ; ; t -:.:; n_max ; s (*square(*pi/*(t+/4))+)/; figure() plot(t, s); grid axis([ -..]); he matlab code for plotting and evaluating the Fourier Series Expansion his code builds the series incrementally using the for loop Make sure you study this code!! s_e /*ones(size(t)); for n::n_max s_e s_e + (-)^((n-)/) * */(n*pi) * cos(*pi*n/*t); end figure() plot(t, s,'b-', t, s_e,'r--'); axis([ -..]); legend('original s(t)', 'up to n '); grid /7/6 Dr. shraf S. Hasan Mahmoud 4

22 Notes Previous Example he more terms included in the series expansion the closer the representation to the original s(t) i.e. comparing s(t) with s_e(nn*), the greater the n* the closer the representation is How to measure closeness? nswer: Let s use power!! /7/6 Dr. shraf S. Hasan Mahmoud 43 Power Calculation Using Fourier Series Expansion Rule: if s(t) is represented using Fourier Series expansion, then its power can be calculated using: P s s( t) dt 4 + [ n + Bn ] n + C n 4 n /7/6 Dr. shraf S. Hasan Mahmoud 44

23 Power Calculation Using Fourier Series Expansion () he previous result is based on the following two facts: () For f(t) constant power of f(t) constant Proof: power / X constant dt / X constant X constant Watts /7/6 Dr. shraf S. Hasan Mahmoud 45 Power Calculation Using Fourier Series Expansion (3) he previous result is based on the following facts (continued): () For f(t) cos(πnf t+θ) power of f(t) / Proof: Pf f ( t) dt cos (πnft + [ + cos(4πnft + θ )] dt [ + ] θ ) dt /7/6 Dr. shraf S. Hasan Mahmoud 46 3

24 Example : Problem: What is the power of the signal s(t) used in previous example? nd find n* such that the power contained in s_e(nn*) is 95% of that existing in s(t)? Solution: Let the power of s(t) be given by P s s s( t) dt P.5 /7/6 Dr. shraf S. Hasan Mahmoud 47 Example : cont d Now it is desired to compute the power using the Fourier Series Expansion What is the power in s_e(n) /? ns: we apply the power formula: P s _ e( n ) s _ e( n ) 4 4 /7/6 Dr. shraf S. Hasan Mahmoud 48 dt.5 4

25 Example : cont d What is the power in s_e(n) / + /π cos(πf t) ns: we can use the result on slide Power Calculation Using Fourier Series Expansion: P s _ e( n ) s _ e( n ) dt π. 456 π /7/6 Dr. shraf S. Hasan Mahmoud 49 Example : cont d What is the power in s_e(n3) / + /π cos(πf t) /(3π) cos(π3f t) ns: we can use the result on slide Power Calculation Using Fourier Series Expansion: Ps _ e( n 3) s _ e( n 3) dt π 9π + 4 π 9π /7/6 Dr. shraf S. Hasan Mahmoud 5 5

26 Example : cont d What is the power in s_e(n5) / + /π cos(πf t) /(3π) cos(π3f t) + /(5π) cos(π5f t) ns: we can use the result on slide Power Calculation Using Fourier Series Expansion: P s _ e( n 5) s _ e( n 5) dt π 9π + 4 π 9π 5π π /7/6 Dr. shraf S. Hasan Mahmoud 5 Example : cont d What is the power in ( n ) / ( ) s _ e( n ) + cos(πnft) π n,3,5 n ns: we can use the result on slide Power Calculation Using Fourier Series Expansion: P s _ e( n ) s _ e( n ) dt π. 5 n n π his the EXC SME power contained in s(t) his is expected since s(t) is % represented by s_e(n ) /7/6 Dr. shraf S. Hasan Mahmoud 5 n n 6

27 Example : cont d s_e(nk) Expression Power % Power + k /.5 (.5 )/(.5 ) 5% k / + /πcos(πf t).456 (.456 )/(.5 ) 9.5% k * / + /πcos(πf t) % k 3 / + /πcos(πf t) /(3π)cos(π3f t) k 5 / + /πcos(πf t) /(3π)cos(π3f t) + /(5π)cos(π5f t) % % + % power power of s_e(nk) relative to original power in s(t) which is equal to.5 /7/6 Dr. shraf S. Hasan Mahmoud * 53 For k, the expression s_e(nk) is the same as that for s_e(k). Why? Example : cont d herefore, s_e(nn*) such that 95% of power is contained n* 3 /7/6 Dr. shraf S. Hasan Mahmoud 54 7

28 Power Spectral Density Function Fourier Series Expansion: Specifies all the basic harmonics contained in the original function s(t) C n / ( n +B n ) / determines the power contribution of the nth harmonic with frequency nf he power Spectral Density function is a function specifying: how much power is contributed by a given frequency /7/6 Dr. shraf S. Hasan Mahmoud 55 Power Spectral Density Function () ypical PSD function for periodic signals: PSD(f) C / s( t) Periodic s(t) () Fourier Series Expansion + Cn cos(πnft + θn) n () Power calculation ( /) C / Ps s( t) dt + 4 n C n f f C 3 / C 4 / (3) Plotting power versus frequency frequency f f 3f 4f /7/6 Dr. shraf S. Hasan Mahmoud 56 8

29 Power Spectral Density Function (3) mathematical expression for PSD(f) can be written as f PSD( f ) Cn / f n f otherwise nother way (more compact) of writing PSD(f) is as follows: PSD( f ) δ ( f ) + Cn δ ( f nf) 4 where δ(t) is defined by δ ( f ) f f /7/6 Dr. shraf S. Hasan Mahmoud 57 n Power Spectral Density Function (4) δ(f) is referred to as the dirac function or unit impulse function δ(f) f /7/6 Dr. shraf S. Hasan Mahmoud 58 9

30 Note on the PSD Function PSD function has units of Watts/Hz For periodic signals PSD is a discrete function - defined for integer multiples of the fundamental frequency Specifies the power contribution of every harmonic component C n / nf he separation between the discrete components is at least f It is exactly f if all C n s are not zeros E.g. for the previous s(t) example, C n for even n separation f /7/6 Dr. shraf S. Hasan Mahmoud 59 Note on the PSD Function () o calculate the total power of signal Integrate PSD over all contained frequencies For discrete PSD: integration summation herefore total power of s(t), P s ( n /) + C n / in Watts /7/6 Dr. shraf S. Hasan Mahmoud 6 3

31 Example 3: Find the PSD function of the periodic signal s(t) considered in Example. From Example, s(t) is given by s t) + π ( ) n ( n ) ( n,3,5 cos(πnf t) Using Example : Power at the zero frequency (/) /4 Power at the nth harmonic (n odd) is equal to /(nπ) Power at the nth harmonic (n even) is zero herefore the PSD function is given by PSD( f ) 4 δ ( f ) + π /7/6 Dr. shraf S. Hasan Mahmoud 6 / n,3,5 δ ( f nf ) n Example 3: cont d he PSD is plotted as shown (, ) PSD( f ) δ ( f ) + 4 π n,3,5 δ ( f nf ) n.5. Power Spectral Density function for s(t) -, frequency - Hz /7/6 Dr. shraf S. Hasan Mahmoud 6 3

32 Example 3: cont d Matlab Code to plot PSD clear all ; ; t -:.:; n_max ; Frequency [::n_max]; PwrSepctralD zeros(size(frequency)); % Record the DC term power at f PwrSepctralD() (/)^; % Record the nth harmonic power at f nf for n::n_max PwrSepctralD(n+) (*/(n*pi))^ / ; end figure() stem(frequency, PwrSepctralD,'rx'); title('power Spectral Density function for s(t) -, '); xlabel('frequency - Hz'); grid he stem function is typically used to plot discrete functions /7/6 Dr. shraf S. Hasan Mahmoud 63 Example 4: his is a typical exam question Problem: Consider the periodic half-wave rectified signal s(t) depicted in figure. Write a mathematical expression for s(t) Calculate the Fourier Series Expansion for s(t) Calculate the total power for s(t) Find n* such that s_e(n*) has 95% of the total power Determine the PSD function for s(t) Plot the PSD function for s(t) s(t) - /7/6 Dr. shraf S. Hasan Mahmoud 64 t 3

33 Example 4: cont d nswer: (a) o write a mathematical expression for s(t), remember that the general form of a sinusoidal function is given by cos(π X Freq X t), or cos(π /Period X t) herefore s(t) is given by s(t) cos(π/ t) -/4 < t /4 /4 < t 3/4 /7/6 Dr. shraf S. Hasan Mahmoud 65 Example 4: cont d nswer: (b) he F.S.E of s(t): he DC term is given by s( t) dt cos(πt / ) dt sin(πt / ) π π s( t) π [ cos(πnf t) + B sin(πnf t ] + ) n n n [ sin( π / ) sin( / ) ] t π t /7/6 Dr. shraf S. Hasan Mahmoud 66 33

34 Example 4: cont d Remember: sin(ax+bx) sin(ax-bx) [cos(ax)cos(bx)] dx (a+b) (a-b) for a b sin(a+/-b) sin(a)cos(b) +/- cos(a)sin(b) nswer: he n term is given by (remember / f ) n s( t)cos(πnf t) dt cos(πt / )cos(πnft) dt sin 4 cos π ( π ( n + ) ft) π ( n + ) f ( nπ / ) ( n + ) cos + sin + 4 ( nπ / ) ( n ) ( π ( n ) ft) π ( n ) f t t /7/6 Dr. shraf S. Hasan Mahmoud his means: the n should be 67 special! For n For n Example 4: cont d But cos(nπ/) n odd, n (-) (+n/) n even herefore n ( π (+ n / ) (+ n / ) ) ( )( ) + ( n + ) ( n ) For n even For n odd, n /7/6 Dr. shraf S. Hasan Mahmoud 68 34

35 Example 4: cont d he expression for n (for even n) can be further simplified to n ( π ( π ( ) π n ( ) ( ) π (+ n / ) (+ n / ) ) ( )( ) + ( n + ) ( n ) (+ n / ) (+ n / ) ) ( n ) + ( )( ) ( n + ) ( n + )( n ) (+ n / ) (+ n / ) [ ( n ) ( ) ( n ) ] + (+ n / ) ( n ) For n even /7/6 Dr. shraf S. Hasan Mahmoud 69 Example 4: cont d n is still not completely specified we still need to calculate it for n; in other words we need to calculate : n s( t)cos(π ft) dt cos(πt / )cos(πf t) herefore: cos (πf t) dt t + sin 4 πf ( 4πf t) t sin + 4 /7/6 Dr. shraf S. Hasan Mahmoud 7 dt ( π ) sin( π ) 8πf t 35

36 Example 4: cont d his mean n is equal to the following: n /π n n odd, n / n (-) (+n/) n, 4, 6, π(n -) he above expression specifies n for LL POSSIBLE values of n specification is complete /7/6 Dr. shraf S. Hasan Mahmoud 7 Example 4: cont d We still need to compute B n : B n cos 4 Remember: - cos(ax+bx) cos(ax-bx) [sin(ax)cos(bx)] dx (a+b) (a-b) for a b s( t)sin(πnf t) dt cos(πt / )sin(πnft) dt ( π ( n + ) ft) π ( n + ) f cos 4 ( π / ( n + ) ) + cos( π / ( n + ) ) ( n + ) ( π ( n ) ft) π ( n ) f t cos cos π n t For n ( π / ( n ) ) cos( π / ( n ) ) ( ) For n /7/6 Dr. shraf S. Hasan Mahmoud his means: the n should be 7 special! 36

37 Remember: Example 4: cont d sin(ax) cos(ax) sin(ax) B B n is still NO completely specified we still need to calculate it for n; in other words we need to calculate B : n s( t)sin(π ft) dt cos(πt / )sin(πf t) herefore: B cos(πf t)sin(πf t) dt sin(4πf t) t cos(4πf t) π t 4π 4π [ cos( π ) cos( )] his means B n for all n /7/6 Dr. shraf S. Hasan Mahmoud 73 dt dt Example 4: cont d o summarize: n /π n n odd, n / n nd (-) (+n/) n, 4, 6, π(n -) B n for all n Having computed n and B n we are now in a position to write the Fourier Series Expansion for s(t) /7/6 Dr. shraf S. Hasan Mahmoud 74 37

38 Example 4: cont d he Fourier Series Expansion for s(t) is given by s( t) [ cos(πnf t) + B sin(πnf t ] + ) n n n + π ( ) (+ n / ) cos(πf t) + cos(πnf t) π n,4,6 n he C n terms (there is a typo in the textbook) are as follows: C /π C / (-) (+n/) C n , n, 4, 6, π(n ) /7/6 Dr. shraf S. Hasan Mahmoud n odd, n 75 Remember: Example 4: cont d sin(ax) ½ cos(ax) sin(ax) Plot for s(t) and the Fourier Series Expansion for k,,, and 4. Original signal and its Fourier Series Expansion orignal s(t) s e (k) s e (k) s e (k) s e (k4).8 Note: s k increases s_e(nk) approaches the original s(t) mplitude ime (sec) /7/6 Dr. shraf S. Hasan Mahmoud 76 38

39 Example 4: cont d he total power of s(t) is given by: P s 3 s( t) dt t sin(4πt / ) + 8 t / π cos (πt / ) t t 4 herefore total power of s(t).5 /7/6 Dr. shraf S. Hasan Mahmoud 77 Example 4: cont d o find n* such that power of s_e(nn*) 95% of total power: s_e(nk) Expression Power % Power + k /π.3 (.3 )/(.5 ) 4.5% k /π + / cos(πf t).63 (.6 )/(.5 ) 9.5% k /π + / cos(πf t) + /(3π) cos(πf t).488 (.488 )/(.5 ) 99.5% herefore n* power of s_e(n).488 which is 99.5% of total power of s(t) /7/6 Dr. shraf S. Hasan Mahmoud 78 39

40 Example 4: cont d he PSD function for s(t) is as follows: Power for DC term (/π) Power for harmonic at f f : (/) / /8 Power for harmonic at f nf (n,4,6, ): [/(π(n -))] / /(π(n -)) herefore PSD function equals to PSD( f ) δ ( f ) + π δ ( f 8 f ) + π δ ( f nf ) n,4,6 ( n ) /7/6 Dr. shraf S. Hasan Mahmoud 79 Example 4: cont d Plot of he PSD function for s(t).4.. Power spectral density for s(t) Power - Watts Multiples of fundamental freuqncy f (Hz) /7/6 Dr. shraf S. Hasan Mahmoud 8 4

41 Fourier ransform Fourier Series Expansion analysis is applicable for PERIODIC signals ONLY here are important signals that are not periodic such as Your voice waveform Pulse signal p(t) used for modulation and transmission Examples: p (t) and p (t) p (t) p (t) τ/ τ/ t /7/6 Dr. shraf S. Hasan Mahmoud τ/ τ/ 8 t Fourier ransform () How to find the frequency content of such signals? Use FOURIER RNSFORM X ( f ) πjft x( t) e dt πjft x( t) X ( f ) e df /7/6 Dr. shraf S. Hasan Mahmoud 8 4

42 Notes on Fourier ransform F. describes a two-way transformation x(t) X(f) where x(t) is the time representation of the signal, while X(f) is the frequency representation of the signal X(f) is defined on a continuous range of frequencies ll frequencies within the range of X(f) where X(f) is not zero contribute towards building x(t) /7/6 Dr. shraf S. Hasan Mahmoud 83 Notes on Fourier ransform () he magnitude of the contribution of a particular frequency f* in x(t) is proportional to X(f*) Example: Consider the F.. pair shown below clearly frequencies belonging to (-/τ,/τ) contribute significantly more compared to frequencies belonging to (/τ, ) or (-,-/τ) x(t) X(f) F.. τ/ τ/ t -/τ /τ f /7/6 Dr. shraf S. Hasan Mahmoud 84 4

43 Properties of Fourier ransform If x(t) is time-limited X(f) is not frequency-limited i.e. the range of X(f) (-, ) X(f) is complex-valued (has magnitude and phase) in general, i.e. () C If x(t) is a real-valued symmetric X(f) is real-valued, i.e. () R /7/6 Dr. shraf S. Hasan Mahmoud 85 Relation between Fourier Series Expansion and Fourier ransform Consider the following two signals: s (t) x(t) is the same as s(t) except that its period x(t) -/4 /4 3/4 5/4 t τ/ τ/ t S(f) For S(f) Separation between the discrete frequencies equal to f or / X(f) -3/ -/ / -/τ /τ 3/ f f -4/ -/ 4/ Separation become zero as and the / discrete function S(f) continuous function X(f) /7/6 Dr. shraf S. Hasan Mahmoud 86 43

44 Relation between Fourier Series Expansion and Fourier ransform () he separation between spectral lines for a periodic signal is / s infinity and s(t) becomes non periodic the separation between spectral lines zero (i.e. it becomes continuous) /7/6 Dr. shraf S. Hasan Mahmoud 87 Example 5: Problem: Consider the square pulse function shown in figure: Write a mathematical expression for p(t) Find the Fourier transform for p(t) Plot P(f) p(t) -τ/ τ/ t /7/6 Dr. shraf S. Hasan Mahmoud 88 44

45 Example 5: cont d nswer: p(t) can be expressed as p(t) t τ/ otherwise he F.. for p(t), P(f) is given by P( f ) πjft p( t) e dt /7/6 Dr. shraf S. Hasan Mahmoud 89 P(f) is real-valued for this specific example of p(t); why? Example 5: cont d Which is equal to P( f ) p( t) e dt πjf πf τ πjft πjft e τ τ πjft e dt τ πjfτ πjfτ ( e e ) j sin( πfτ ) τ πfτ /7/6 Dr. shraf S. Hasan Mahmoud 9 dt πjf πjfτ πjfτ ( e e ) Remember: Euler identity : e jx cos(x) + j sin(x), OR cos(x) (e jx +e -jx )/ sin(x) (e jx -e -jx )/(j) 45

46 Example 5: cont d P(f) plot for and τ Note: P(f) is define on (-, ) P(f) is continuous (except at f Hz) P(f) ZERO for f n/τ Fourier ransform of p(t) -, aw For practical pulses P(f) approaches zero as f ±. Most of the energy of p(t) is contained in the period of (-/τ, /τ) frequency - Hz /7/6 Dr. shraf S. Hasan Mahmoud 9 Energy Spectral Density Function (ESDF) ESDF is defined as () () () where P(f) is the F.. of the pulse p(t). P*(f) is the complex conjugate of P(f). ESDF is a measure of how much energy is contained at a particular frequency f Units of ESDF is Joules per Hz How would you compare ESDF with PSDF? /7/6 Dr. shraf S. Hasan Mahmoud 9 46

47 Example 5 (For Volts and aw sec) - cont d.8 (a) pulse (time domain).8 (b) Magnitude of F. (freq. domain) amplide (volts).6.4. abs(p(f)) time - (sec) -4-4 frequency - (Hz) < () otherwise () () (a) sin() () () () sin() -4-4 /7/6 Dr. shraf S. Hasan Mahmoud frequency - (Hz) 93 (b) (c) ESDP (Joules/Hz) (c) ESDF (freq. domain) Example 5 (For Volts and aw sec) Matlab Code Code for producing plots on previous slide clear all; LineWidth 3; FontSize 4; %Example of rectangular pulse ; aw ; % parameters for the rectangle pulse t_step.; f_step.; Nmax 4; t -aw:t_step:aw; % define the time axis f -Nmax/aw:f_step:Nmax/aw; % define the frequency axis p_t *rectpuls(t/aw); % he rectangle pulse of height and width aw P_f *aw*sin(pi*aw*f)./(pi*aw*f); % he corresponding F. P(f) ESDF_f P_f.*conj(P_f)/(*pi); % he ESDF figure(); clf; set(gca, 'FontSize', FontSize); h plot(t, p_t, '-r', 'LineWidth', LineWidth); xlabel('time - (sec)'); ylabel('amplide (volts)'); grid on; figure(); clf; set(gca, 'FontSize', FontSize); h plot(f, abs(p_f), '-r', 'LineWidth', LineWidth); xlabel('frequency - (Hz)'); ylabel('abs(p(f))'); grid on; figure(3); clf; set(gca, 'FontSize', FontSize); h plot(f, ESDF_f, '-r', 'LineWidth', LineWidth); xlabel('frequency - (Hz)'); ylabel('esdp (Joules/Hz)'); grid on; /7/6 Dr. shraf S. Hasan Mahmoud 94 47

48 Example 6: Repeat example 5 for triangular pulse shown in textbook Figure. page 839. Let be equal to seconds and be equal to 3 volts. /7/6 Dr. shraf S. Hasan Mahmoud 95 Example 7: Problem: If the FSE expansion for the function g(t) is as given below. Compute the FSE for the function s(t) without computing the FSE coefficients for s(t). g(t) s(t) -Τ/ Τ/ t -Τ/ Τ/ t /7/6 Dr. shraf S. Hasan Mahmoud 48

49 Z-ransform Digital signals Discrete-time signals sampled continues-time signals E.g. I/O signal used by micro-controllers Systems described by difference equations E.g. CD player contains digital signal processing system (digital filter) to manipulate the digital audio signal For such systems input and output are related by difference equations and the Z- transform is used to solve these systems /7/6 Dr. shraf S. Hasan Mahmoud 97 Z-ransform - Introduction For Modern systems analog signal /D samples Signal processor samples D/ analog output processing filtering noise, equalizing music, adding effects to audio/video, etc. /7/6 Dr. shraf S. Hasan Mahmoud 98 49

50 Z-ransform - Definition Let y(t) be a continuous-time signal (function) Define sampling period hen y(k) y k y(k) for k,,, 3, defines uniformly spaced samples of the original signal y(t) he Z-ransform for yk is defined as () if the summation converges /7/6 Dr. shraf S. Hasan Mahmoud 99 Z-ransform Example Compute the Z-transform for the sequence () for k,,, and <a< Compute the Z-transform for the sequence plot for y(k) is shown for < a < Note the y(k) is defined only on specific indices (time samples) pplying the definition () for a/z < sample index (k) /7/6 Dr. shraf S. Hasan Mahmoud for k,,, signal y(k) ( ) y and a.5 pair () 5

51 Z-ransform Example Compute the Z-transform for the sequence () (.9) for k,,, and <a< Compute the Z-transform for the sequence plot for y(k) is shown for < a < Using the previous result for.9/z <. () signal y(k) y and a sample index (k) /7/6 Dr. shraf S. Hasan Mahmoud Z-ransform Example 3 Compute the Z-transform for the sequence () Compute the Z-transform for the sequence Using the definition for all z. Δ() -5 5 sample index (k) /7/6 Dr. shraf S. Hasan Mahmoud signal δ(k) () () Discrete-time Delta Function () pair () 5

52 Z-ransform Example 3 Compute the Z-transform for the sequence (),,, otherwise Compute the Z-transform for the sequence Using the definition Discrete-time Unit-Step Function signal δ(k) sample index (k) () () for /z <. () pair (3) /7/6 Dr. shraf S. Hasan Mahmoud 3 Z-ransform Example 4 Compute the Z-transform for the sequence Discrete-time arbitrary function 6 5 y(k) [, 4, 6, 4,, ] for k,,, 3, 4, and 5, respectively Compute the Z-transform for the sequence signal δ(k) 4 3 Using the definition () for /z <. () sample index (k) /7/6 Dr. shraf S. Hasan Mahmoud 4 5

53 Inverse Z-ransform Example 5 Compute the Inverse Z-ransform for We use pair () y and a 3 Or y(k) is given by for k,,, () (.3).3 y(k) may be rewritten as (.3) () /7/6 Dr. shraf S. Hasan Mahmoud 5 Inverse Z-ransform Example 6 Compute the Inverse Z-ransform for Solution? () (.3)(.9) /7/6 Dr. shraf S. Hasan Mahmoud 6 53