Fourier Transform for Continuous Functions Central goal: representing a signal by a set of orthogonal bases that are corresponding to frequencies or spectrum. Fourier series allows to find the spectrum of periodic functions. How to define the frequency or spectrum for general continuous-time signals?
From Periodic to Non-Periodic Fourier series transforms a periodic continuous signal into the frequency domain. What will happen when the continuous signal is not periodic? Consider the period of a signal with the fundamental frequency 0. T 0 specifies the fundamental period, T 0 2 1 w f 0
Review of Fourier Series Fourier series representation of a periodic signal x T0 (t) can be given by the pair of equations Forward Transform Integrals over a period [0,T 0 ] and [-T 0 /2, T 0 /2] are the same Inverse Transform where ω 0 = 2π T 0
Imaging T 0 A non-periodic signal can be conceptually thought of as a periodic signal whose fundamental period T 0 is infinite long, T 0. In this case, the fundamental frequency 0 0. Remember that the spectrum (in the frequency domain) of a periodic continuous signal is discrete, specified by k 0 (k is an integer). Therefore, the interval between adjacent frequencies is just 0.
Extreme Case of Fourier Series: T 0 Let us re-denote f(t) = x T0 t, and rewrite the forward transform of Fourier series as 1 T 0 jkw0t a x t e dt k T0 T 0 /2 0 T /2 w 0 2 / w 0 / w 0 0 f t e dt jkw t
Extreme Case of Fourier Series Further changing the notation: Let b kω0 = a k, the forward transform becomes w / w 0 0 jkw0t b f t e dt k 0 2 / w The inverse transform becomes 0 f () t b e k k 0 jkw t 0
Extreme Case of Fourier Series When 0 0, we can image that the frequency becomes continuous: The interval 0 becomes d Let = k 0. Then, when 0 0, the forward transform approaches dw ( ) jw t b w f t e dt 2
Extreme Case of Fourier Series Combining with the inverse transform, we have f ( t ) b( w ) e k dw jw t f t f t e dt e 2 b(ω) when 0 0, where the summation in the inverse transform becomes integral. jw t jw t
It becomes: Continuous Fourier Transform Consider the above recovering equation of f(t). Let us decompose the equation into forward transform and inverse transform: 1 jw t jw t f t f t e dt e dw 2 Forward Transform of f(t) to the frequency domain F(w) Inverse Transform of F(w) to the time domain f(t)
Continuous Fourier Transform a.k.a. Continuous-time Fourier Transform Forward Transform F Inverse Transform f jw f t 1 Remark: in the continuous domain, e jt (R) still form a set of orthogonal bases, no matter whether is a multiple of an integer or not. e t F jw 2 jwt e dt jwt dw
Continuous Fourier Transform Both time and frequency domains in continuous Fourier transform are continuous. The frequency waveform is also referred to as the spectrum. Continuous Fourier transform is the most general Fourier transform. We will see later that other Fourier transforms (including Fourier series) are its special cases. It is also the most standard Fourier transform. When there is no specification, we usually referred to Fourier Transform as this type.
Illustration Example: from Fourier Series to Continuous Fourier Transform We present an example to illustrate the approaching from Fourier series to continuous F. T. Consider a finite-duration signal x(t) that is nonperiodic. Therefore, it does not have a Fourier series representation. Despite using a finite-duration signal as an example for illustration in the following, the argument in previous slides is not restricted to the case of finite-duration signals.
Illustration Example: from Fourier Series to Continuous Fourier Transform Let us use x(t) to define one period of a periodic function as long as the period (T 0 ) is longer than the duration, T 0 >T. We can form a periodic function with period T 0 by repeating copies of x(t) every T 0 seconds.
Illustration Example: from Fourier Series to Continuous Fourier Transform A convenient way to express this periodic replication process is to write an infinite sum of time-shifted copies of x(t) From the figure illustration, it can be seen that the periodical signal becomes the original x(t) when T 0.
Illustration Example: from Fourier Series to Continuous Fourier Transform By forward transform of the Fourier series, the integral of a single period T 0 is as follows:
Spectra by Fourier series of the periodic signal formed by the finite duration signal x(t). (a) T 0 = 2T (b)t 0 = 4T (c) T 0 = 8T.
When T 0, Illustration Example This is just the continuous F. T. of the finite duration signal x(t), Note that the function of the form sin(x)/ x or sin(x)/ ( x) is called the sinc function, which plays an important role in sampling a continuous signal which will be introduced later.
Continuous Fourier Transform Pair Transform pair F Forward jw f t Sometimes also written as e Backward 1 jwt dt f t F jw e 2 jwt dw F jw 1 f t e dt 2 jw t depending on how we decompose the normalization constant 1/2. 1 jw t f t F jw e dw 2
There are Many Variations of the Forms of Continuous F. T. (from Kuhn s slides 2005)
Symmetry between Time and Frequency of Continuous Fourier Transform Unlike Fourier series, the continuous Fourier transform has very similar forward and inverse transforms. Except to the normalization constant, the only difference is that the forward uses j and the inverse uses j in the complex exponential basis. This suggests that the roles of time and frequency can be exchanged, and some properties are symmetric to each other.
Existence and Convergence of Fourier Transform We have derived continuous Fourier transform as an extreme extension of Fourier series. To ensure the existence of continuous Fourier transform, we should consider the conditions whhere the integrals exist. A sufficient condition is
Rational Hence, if x t dt <, or equivalently, the integral is bounded, then the continuous Fourier transform is also bounded.
Existence and Convergence of Fourier Transform The above is only a sufficient condition but not a necessary condition. We will see many examples of functions that do not satisfy the above condition for which we can still obtain a useful Fourier transform representation. Particularly, if we are willing to allow impulse signals (which are not well-behaved functions in the time and frequency domain representation). The impulse signals will be introduced later. In engineering, we usually do not care much about the exact necessary and sufficient conditions, despite there are mathematically rigorous ways to specify these conditions.
What is the continuous Fourier transform of a periodic signal? In the above, we extend the Fourier series when the period T 0 for a periodical signal. What happens when we substitute a periodic signal into the continuous Fourier transform? By doing so, assume that f(t) is a periodic signal with period T 0. From Fourier series, we know that f(t) can be represented as jw t F jw f t e dt jkw 0 t, f t a e w 0 k k 2 T 0
Preliminary Property When 0, the integral of e jt over the whole range is zero: e jw t dt 0 It is because that jw t k ( 2 / w)( k1) ( 2 / w) k jw t e dt e dt k 1 jw e jw t t ( 2 / w )( k 1) t ( 2 / w ) k 0
What is the continuous Fourier transform of a periodic signal? The continuous Fourier transform of a periodic signal becomes 0 k k jkw t jkw t jw t j ( kw w ) t 0 0 a e e dt a e dt k k k 0, w kw 0 0, w a 1 dt, w kw, w k 0 jw t F jw a e e dt k kw kw 0 0
Continuous Fourier Transform of a Periodic Signal That is, we can see that when is a integer multiple of the fundamental frequency 0 (i.e., =k 0, k is an integer), the continuous F. T. becomes infinity. Rigorously speaking, the continuous F. T. doesn t exist. So, what can we do? Should we separate the signals into two types, one is periodic and the other is non-periodic? When we encounter a periodic signal, are we only allowed to use Fourier series? Moreover, even for very simple signals such as f(t)=1, its continuous F. T. doesn t exist either according to the same argument.
Impulse Function To overcome this difficulty, and also to make continuous F. T. more applicable, the Dirac s delta function (or unit impulse function) is introduced. The impulse time-domain signal is the most concentrated time signal that we can have. Using an arrow to indicate it
What is it mathematically? Dirac s delta function can be thought of as the limit of function sequence such as The delta (or unit impulse) function is mathematically speaking not a function, but a distribution, that is in an expression that is only defined when integrated. In engineering, we don t care about many of the possible ways for its rigorous definition. We only care its property when computing integrals.
Some properties of Delta (Unit Impulse) (1) Function (2) So, the Fourier transform pair of delta function is
Further properties Since the time domain and the frequency domain are symmetric in continuous Fourier transform, we also have jw t e dt ( w ) Another property: the integral of unit impulse over the whole range is 1:
Continuous Fourier Transform of a Periodic Signal By introducing the impulse function, let s go back to see the continuous Fourier transform of a periodic signal. Now, 0 jkw t j ( kw0 w ) t a e dt a ( w kw ) k k 0 k k Remark: the unit-impulse function is allowed to be multiplied by a constant (here, a k ) to reflect the magnitude of an impulse. k jw t F jw a e e dt k
Continuous Fourier Transform of a Periodic Signal Hence, the continuous Fourier transform of a periodic signal is an impulse train, with each unit impulse weighted by a k. F(jw) a 0 a 1 a 2 a 3 0 w 0 2w 0 3w 0 w Note that a k is just the spectrum of Fourier series. Hence, the magnitudes of the impulse functions are proportional to those computed from the discrete Fourier series.
Continuous Fourier Transform A general Fourier Transform for Spectrum Representation With the unit-impulse function incorporated, the continuous Fourier transform can represent a broad range of continuous-time signals. It is the most general F. T. as mentioned before, including Fourier series as its special case. It can also take discrete-time signal as input. We will investigate it later. It provides a unified and general definition of spectrum. When mention the spectrum of a signal, it usually means the continuous Fourier transform of the signal.
Examples of Fourier Transform Pairs Rectangular function (rectangular pulse signal) Derivation of its continuous F. T.
Fourier transform of rectangular function Rectangular function can also be represented by the unit-pulse function u(t) as T T u ( t ) u ( t ) 2 2 where the unit-step function is u() t 1, t 0 0, t 0 Hence, we have the Fourier transform pair: A real function in frequency domain
Time and Frequency domains are dual
Fourier transform of right-sided real-exponential signal A complex function in frequency domain Since The real and imaginary parts are
Impulse in Time and Frequency Derivation:
Impulse in Time and Frequency By the duality (or symmetry) between time and frequency domain Intuitive interpretation: the constant signal x(t)=1 for all t has only one frequency, namely DC, and we see that its transform is an impulse concentrated at =0.
Complex Exponential It says that a complex exponential signal of frequency 0 has a Fourier transform that is nonzero only the frequency 0. Linear Property:
Sinusoids Derivation: Since By the linear property, we have
Periodic Signals Example: time domain a periodic square wave
Frequency domain of squared wave
Impulse Train Derivation: because p(t) is a periodic signal, it can be expressed by Fourier series: where
Impulse Train To determine the coefficients a k of Fourier series, we must evaluate the Fourier series integral over one period, i.e.,
Impulse Train Since the continuous Fourier transform of a periodic signal is in general of the form of sum of delta functions centered at integer multiples of s : Hence, the Fourier transform of the impulse train p(t) is another impulse train
Properties of Fourier Transform Pairs Scaling Property
Properties of Fourier Transform Pairs Flip Property Derivation: from the scaling property, we have
Properties of Fourier Transform Pairs Time delay property Differentiation property
Symmetry Properties of Fourier Transform Pairs If we take complex conjugate of the spectrum, we obtain Hence, X*(-jw) is the Fourier transform of x*(t)
Symmetry Properties of Fourier Transform Pairs Therefore, if x(t) is a real function, x(t)=x*(t), the above property reveals that X(jw) = X*(-jw). Hence, we have Then, we can conduct that Re{X(jw)} is an even function Im{X(jw)} is a even function
Symmetry Properties of Fourier Transform Pairs In other words, when x(t) is real, the real part of its Fourier transform X(jw) is even, and the imaginary part is odd. Similarly, we also have a symmetric property for magnitude (amplitude) and phase in polar form when x(t) is real That is, the magnitude spectrum of real signal is even, and the phase spectrum is odd.
Example: magnitude and phase spectra of a real signal Even function Odd function
Summary of Fourier transform symmetries
Multiplication in the frequency domain When multiplying two signals in the frequency domain, what will be obtained in the time domain? Convolution: a moving average operation: Convolution of two continuous-time signals x(t) and h(t) is defined as t x ht It can be written in short by Convolution is commutative y d ( ) ( ) y t x t h t x ( t ) h ( t ) x h t d h s x t s ds h ( t ) x ( t )
Illustration example of convolution
Animation example of convolution
Animation example of convolution
From Kuhn 2005 2D continuous convolution: optics example
Convolution in time domain Derivation
Convolution Property Interchange the order of integrals Let
Convolution Property By substitution back, Convolution property is one of the most important properties in Fourier transform.
Convolution Property Due to the duality of frequency and time domains, we also have the property that multiplication in the time domain corresponds to the convolution in the frequency domain: That is
Example: AM Time domain multiplication Frequency domain convolution
Basic Fourier Transform Pairs
Basic Fourier Transform Pairs
Basic Fourier Transform Properties
Basic Fourier Transform Properties
Fourier Transform of Discrete-time Signals From Fourier Series: time domain is periodic, then frequency domain is discrete Remember that the continuous Fourier transform has similar forms of forward and inverse transforms. Question: When time domain is discrete, what happens in the frequency domain? The frequency domain is periodic Discrete time Fourier transform: dealing with the case where time domain is a discrete-time signal, x[-2t], x[-t], x[0], x[t], x[2t], (T is the time step)