Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., x(t T ) x(t ), t R Example: the rectangular pulse train The Fourier Series Then, x(t) can be expressed as x(t ) ck e jk 0t
, t k where 0 2 / T is the fundamental frequency (rad/sec) of the signal and T /2 1 jk ot ck x(t )e dt , k 0, 1, 2, T T /2 c0 is called the constant or dc component of x(t) Dirichlet Conditions
A periodic signal x(t), has a Fourier series if it satisfies the following conditions: 1. x(t) is absolutely integrable over any period, namely a T | x(t ) | dt , a a 2. x(t) has only a finite number of maxima and minima over any period 3. x(t) has only a finite number of
discontinuities over any period Example: The Rectangular Pulse Train From figure T 2, so 0 2 / 2 Clearly x(t) satisfies the Dirichlet conditions and thus has a Fourier series representation Example: The Rectangular Pulse Train Contd k jkt 1 1
x(t) sin e , t 2 k k 2 k0 Trigonometric Fourier Series By using Eulers formula, we can rewrite as x(t ) ck e jk 0 t , t
k x(t ) c0 2 | ck |cos(k 0t ck ), t k 1 dc component k-th harmonic as long as x(t) is real This expression is called the trigonometric Fourier series of x(t)
Example: Trigonometric Fourier Series of the Rectangular Pulse Train The expression k jkt 1 1 x(t) sin e , t 2 k k 2 k0 can be rewritten as 1 2 ( k 1) / 2
x(t ) cos k t ( 1) 1 , t 2 k 1 k 2 k odd Gibbs Phenomenon Given an odd positive integer N, define the N-th partial sum of the previous series 1 xN (t ) 2 N
k 1 k odd 2 ( k 1) / 2 cos k t ( 1) 1 , t k 2 According to Fouriers theorem, theorem it should be
lim | xN (t ) x(t ) |0 N Gibbs Phenomenon Contd x3 (t ) x9 (t ) Gibbs Phenomenon Contd x21 (t ) x45 (t )
overshoot: overshoot about 9 % of the signal magnitude (present even if N ) Parsevals Theorem Let x(t) be a periodic signal with period T The average power P of the signal is defined as T /2 1 2 P x (t )dt T T /2
jk 0t x ( t ) c e Expressing the signal as k , t it is also k
2 P | ck | k Fourier Transform We have seen that periodic signals can be represented with the Fourier series Can aperiodic signals be analyzed in terms of frequency components? Yes, and the Fourier transform provides the tool for this analysis The major difference w.r.t. the line spectra of periodic signals is that the spectra of aperiodic signals are defined for all real values of the frequency variable not just for a discrete set of values
Frequency Content of the Rectangular Pulse x(t ) xT (t ) x(t ) lim xT (t ) T Frequency Content of the Rectangular Pulse Contd Since xT (t ) is periodic with period T, we can write
xT (t ) ck e jk 0t , t k where T /2 1 jk o t ck x(t )e dt , k 0, 1, 2, T T /2
Frequency Content of the Rectangular Pulse Contd What happens to the frequency components of xT (t ) as T ? For k 0 : c0 1/ T For k 1, 2, : 2 k0 1 k0 ck sin sin k0T 2 k
2 0 2 / T Frequency Content of the Rectangular Pulse Contd plots of T | ck | vs. k 0 for T 2,5,10 Frequency Content of the Rectangular Pulse Contd It can be easily shown that lim Tck sinc T 2
, where sin( ) sinc( ) Fourier Transform of the Rectangular Pulse The Fourier transform of the rectangular pulse x(t) is defined to be the limit of Tck as T , i.e.,
X ( ) lim Tck sinc T 2 | X ( ) | , arg( X ( )) The Fourier Transform in the General Case Given a signal x(t), its Fourier transform X ( ) is defined as
X ( ) x(t )e j t dt , A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges The Fourier Transform in the General Case Contd The integral does converge if
1. the signal x(t) is well-behaved well-behaved 2. and x(t) is absolutely integrable, integrable namely, | x(t ) | dt 1. Note: well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval Example: The DC or Constant Signal Consider the signal x(t ) 1, t Clearly x(t) does not satisfy the first
requirement since | x(t ) | dt dt Therefore, the constant signal does not have a Fourier transform in the ordinary sense Later on, well see that it has however a Fourier transform in a generalized sense Example: The Exponential Signal bt
Consider the signal x(t ) e u (t ), b Its Fourier transform is given by bt X ( ) e u (t )e j t dt t
e 0 ( b j ) t 1 ( b j ) t dt e b j t 0 Example: The Exponential Signal Contd
If b 0 , X ( ) does not exist If b 0 , x(t ) u (t ) and X ( ) does not exist either in the ordinary sense If b 0 , it is 1 X ( ) b j amplitude spectrum 1 | X ( ) | b2 2 phase spectrum
arg( X ( )) arctan b Example: Amplitude and Phase Spectra of the Exponential Signal x(t ) e 10t u (t ) Rectangular Form of the Fourier Transform Consider X ( ) x(t )e j t
dt , Since X ( ) in general is a complex function, by using Eulers formula X ( ) x(t ) cos( t ) dt j x(t )sin( t ) dt R ( )
X ( ) R ( ) jI ( ) I ( ) Polar Form of the Fourier Transform X ( ) R ( ) jI ( ) can be expressed in a polar form as X ( ) | X ( ) | exp( j arg( X ( ))) where 2 2 | X ( ) | R ( ) I ( ) I ( )
arg( X ( )) arctan R ( ) Fourier Transform of Real-Valued Signals If x(t) is real-valued, it is X ( ) X ( ) Moreover Hermitian symmetry X ( ) | X ( ) | exp( j arg( X ( )))
whence | X ( ) || X ( ) | and arg( X ( )) arg( X ( )) Example: Fourier Transform of the Rectangular Pulse Consider the even signal It is t / 2 2 2 t t / 2 t X ( ) 2 (1) cos( t ) dt sin( t ) t 0 sin
2 0 t t sinc 2 Example: Fourier Transform of the Rectangular Pulse Contd t X ( ) t sinc
2 Example: Fourier Transform of the Rectangular Pulse Contd amplitude spectrum phase spectrum Bandlimited Signals A signal x(t) is said to be bandlimited if its Fourier transform X ( ) is zero for all B where B is some positive number, called the bandwidth of the signal It turns out that any bandlimited signal must
have an infinite duration in time, i.e., bandlimited signals cannot be time limited Bandlimited Signals Contd If a signal x(t) is not bandlimited, it is said to have infinite bandwidth or an infinite spectrum Time-limited signals cannot be bandlimited and thus all time-limited signals have infinite bandwidth However, for any well-behaved signal x(t) X ( ) 0 it can be proven that lim whence it can be assumed that | X ( ) |0 B
B being a convenient large number Inverse Fourier Transform Given a signal x(t) with Fourier transform X ( ) , x(t) can be recomputed from X ( ) by applying the inverse Fourier transform given by 1 x(t ) 2 X ( )e j t
d , t Transform pair x(t ) X ( ) Properties of the Fourier Transform x(t ) X ( ) y (t ) Y ( ) Linearity:
x(t ) y (t ) X ( ) Y ( ) Left or Right Shift in Time: x(t t0 ) X ( )e Time Scaling: j t0 1 x(at ) X a a Properties of the Fourier Transform Time Reversal: x( t ) X ( ) Multiplication by a Power of t:
n d t x(t ) ( j ) X ( ) n d n n Multiplication by a Complex Exponential: x(t )e j 0t
X ( 0 ) Properties of the Fourier Transform Multiplication by a Sinusoid (Modulation): j x(t )sin( 0t ) X ( 0 ) X ( 0 ) 2 1 x(t ) cos( 0t ) X ( 0 ) X ( 0 ) 2 Differentiation in the Time Domain: n d n x(t ) ( j ) X ( )
n dt Properties of the Fourier Transform Integration in the Time Domain: t 1 x(t )dt X ( ) X (0) ( ) j Convolution in the Time Domain: x(t ) y (t ) X ( )Y ( ) Multiplication in the Time Domain:
x(t ) y (t ) X ( ) Y ( ) Properties of the Fourier Transform Parsevals Theorem: 1 x(t ) y (t )dt 2 X ( )Y ( )d
1 2 if y (t ) x (t ) | x (t ) | dt | X ( ) | d 2 2 Duality: X (t ) 2 x( ) Properties of the Fourier Transform Summary
Example: Linearity x(t ) p4 (t ) p2 (t ) 2 X ( ) 4sinc 2sinc Example: Time Shift x(t ) p2 (t 1) j
X ( ) 2sinc e Example: Time Scaling p2 (t ) p2 (2t ) 2sinc sinc 2
a 1 time compression frequency expansion 0 a 1 time expansion frequency compression Example: Multiplication in Time x(t ) tp2 (t ) d d sin cos sin X ( ) j 2sinc j 2
j2 d d 2 Example: Multiplication in Time Contd cos sin X ( ) j 2 2 Example: Multiplication by a Sinusoid
x(t ) pt (t ) cos( 0t ) sinusoidal burst 1 t ( 0 ) t ( 0 ) X ( ) t sinc t sinc 2 2 2
Example: Multiplication by a Sinusoid Contd 1 t ( 0 ) t ( 0 ) X ( ) t sinc t sinc 2 2 2
0 60 rad / sec t 0.5 Example: Integration in the Time Domain 2|t | v(t ) 1 pt (t ) t dv(t ) x(t ) dt
Example: Integration in the Time Domain Contd The Fourier transform of x(t) can be easily found to be t t X ( ) sinc j 2sin 4 4 Now, by using the integration property, it is 1
t 2 t V ( ) X ( ) X (0) ( ) sinc j 2 4 Example: Integration in the Time Domain Contd t 2 t V ( ) sinc 2
4 Generalized Fourier Transform Fourier transform of (t ) (t )e j t dt 1 (t ) 1
Applying the duality property x(t ) 1, t 2 ( ) generalized Fourier transform of the constant signal x(t ) 1, t Generalized Fourier Transform of Sinusoidal Signals cos( 0t ) ( 0 ) ( 0 ) sin( 0t ) j ( 0 ) ( 0 ) Fourier Transform of Periodic Signals Let x(t) be a periodic signal with period T; as such, it can be represented with its
Fourier transform x(t ) ck e k jk 0t 0 2 / T Since e j0t 2 ( 0 ) , it is X ( ) 2 ck ( k 0 ) k Fourier Transform of
the Unit-Step Function Since t u (t ) (t )dt using the integration property, it is t 1 u (t ) (t )dt ( ) j
Common Fourier Transform Pairs