Frequency domain representation of signals
In Topic 4 we saw how to express signals as sums (or integrals for CT signals) of scaled and delayed unit impulse functions. In this topic we discuss a systematic way to represent signals as a sum of scaled complex exponentials (at different frequencies). Thinking about signals as sums of scaled complex exponentials at different frequencies (a so-called frequency-domain representation) is very useful for a number of reasons.
• Frequency domain representations often interpretable and intuitive. In the case of sound, the frequencies of the complex exponential correspond to the pitch (high or low) of the sound. For image patches, high ‘frequencies’ correspond to fast changes in intensity (such as stripes) and low ‘frequencies’ correspond to patches that are fairly constant in intensity.
• Frequency domain representations are often concise. Many real signals can be approxi- mated well by signals that have only a small number of non-zero frequency components. This is one of the ideas behind methods for (lossy) compression of sound and images.
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• It turns out to be easy to understand the effect that an LTI system has on signals, it we represent the signals in terms of their frequency components. We explore this more in the next topic which deals with the frequency response of an LTI system.
5.1 Basic frequency domain signals
In this section we revisit both CT and DT unit complex exponentials and their close friends, sinusoids.
Our aim is to use CT complex exponentials as building blocks to represent more complicated CT signals. In particular we want to represent signals as sums of scaled unit complex exponen- tials. In this section we explore important basic properties of unit complex exponentials, and how they behave under summation and scaling.
5.1.1 Unit complex exponentials in continuous-time
Recall that x(t) = ejω0t is a unit complex exponential. It is periodic with fundamental period T0 = 2π/ω0. The larger the period is, the longer it takes for the signal to complete a ‘cycle’. The signal x(t) has fundamental frequency ω0 rad/s. The larger the frequency is the faster the signal completes a ‘cycle’.
Units of frequency
Most of the time in this unit we describe continuous frequency in terms of radians/second. This unit is most convenient mathematically. It is also very useful to consider the more intuitive unit of Hz (cycles/second). If ω0 is the frequency in radians/second and f0 is the frequency in Hz, then these are related by
|ω0| = 2π|f0| = 2π. T0
The absolute value signs are there because the fundamental period T0 is always positive (by definition) but the fundamental frequency may be positive or negative. There is a nice ‘picture’ that explains these two units and the relationship between them.
Imagine the unit complex exponential x(t) = ejω0t. Think of this as a phasor, i.e., as a rotating arrow in the complex plane that points from the origin to x(t) at each time t. At time t = 0 this phasor is at 1 in the complex plane. As t increases:
• if ω0 is positive then the arrow rotates anti-clockwise around the unit circle at a constant speed;
• if ω0 is negative then the arrow rotates clockwise around the unit circle at a constant speed;
• ifω0 =0thenthearrowstaysat1foralltime.
Whether the frequency is positive or negative tells us the direction of rotation of the phasor.
• The time taken for the arrow to complete one ‘cycle’ around the unit circle is the fundamental period T0 seconds. So the number of cycles per second is f0 = 1/T0.
• A full ‘cycle’ around the unit circle corresponds to the arrow sweeping out an angle of 2π radians. As such, the time take for the arrow to sweep out an angle of 2π radians is the same as the fundamental period T0. So the number of radians swept out per second by the arrow is 2π/T0.
Closely related (via Euler’s formula) to the unit complex exponential are the sine and cosine functions.
Cosine and sine in terms of complex exponentials
Euler’s formula tells us that ejω0t = cos(ω0t) + j sin(ω0t). This can be rearranged to give the following extremely useful expressions for sine and cosine in terms of complex exponentials:
cos(ω0t) = 21 ejω0t + e−jω0t = Re[ejω0t] (5.1)
sin(ω0t) = 1 ejω0t − e−jω0t = Im[ejω0t] (5.2) 2j
If we multiply x(t) = ejω0t by a complex number α the result αejω0t is still periodic with fundamental period T0. What happens if we take the sum of two complex exponentials with different frequencies. When is the sum periodic?
Periodicity of sum of complex exponentials
Let x(t) = αx1(t) + βx2(t) where x1(t) = ejω1t and x2(t) = ejω2t (and α, β ̸= 0). Let T1 = 2π/ω1 be the fundamental period of x1 and let T2 = 2π/ω2 be the fundamental period of x2. The signal x is periodic with period T if and only if there are positive integers k1 and k2 such that
T = k1T1 = k2T2. (5.3) The smallest possible choice of T is the fundamental period, T0.
To see why this is true, note that if x is periodic, any period T must satisfy
x1(t + T) = ejω1(t+T) = ejω1t = x1(t) and x2(t + T) = ejω2(t+T) = ejω2t = x2(t).
For this to be the case we need that T is a valid period of x1 (i.e., an integer multiple of T1) and a valid period for x2 (i.e., an integer multiple of T2). This is exactly the condition described in equation (5.3).
The same sort of argument extends to the problem of determining whether the sum of more than two complex exponentials is periodic, and if so, finding the fundamental period.
Harmonically related frequencies
We say that two frequencies ω1 and ω2 are harmonically related if
ω1 = k1 where k1 and k2 are positive integers
i.e., ω1/ω2 is a rational number. In this case
k 2π = k 2π 1 ω1 2 ω2
and so from equation (5.3) we know that the sum of unit complex exponentials at frequen- cies ω1 and ω2 is periodic whenever the frequencies are harmonically related.
We conclude the section with the fact about complex exponentials that makes Fourier series and Fourier transforms (which we see in due course) work.
Orthogonality of harmonically related complex exponentials
Suppose x1(t) = ejk1ω0t and x2(t) = ejk2ω0t are two complex exponentials at harmonically related frequencies. Let T0 = 2π/ω0 be the fundamental period of any scaled sum of x1 and x2. Then
1T0 1T0 1 ifk1=k2 T x1(t)∗x2(t) dt = T ej(k2−k1)ω0t dt = 0 otherwise.
This integral is easy to check by splitting into the cases where k1 = k2 and k1 ̸= k2. If k1 = k2 then the integral becomes
1 T0 T0 0
If k1 ̸= k2 then the integral becomes
1 T0 j(k2−k1)ω0t 1 j(k2−k1)ω0tT0 ej(k2−k1)ω0T0 − 1
T e dt=j(k−k)ωe 0=j(k−k)ω =0 00 210 210
there the last equality holds because ω0T0 = 2π and (k2 − k1) is an integer so ej(k2−k1)ω0T0 = 1. 5.1.2 Frequency in discrete-time
Recall that x[n] = ejωn is a discrete-time unit complex exponential. We have seen that whether or not x is periodic depends on ω. This is because any period of x must be both
• a valid period of x(t) = ejωt, that is, an integer multiple of 2π/ω, and • a positive integer.
Period and frequency of DT unit complex exponential
A unit DT complex exponential x[n] = ejωn is periodic if and only if there are positive
integers m and N such that
N = m2π. ω
The smallest N for which this is possible is the fundamental period. If N0 is the fundamental frequency then ω0 = 2π/N0 is the fundamental frequency.
The smallest possible fundamental period of a DT signal is N0 = 2. (This is because the constant signal is considered to have frequency 0 and undefined period.) This means there is a maximum possible fundamental frequency of ω0 = 2π/2 = π rad/sample. The prototypi- cal signal with period two samples (the minimum possible) and frequency π rad/sample (the maximum possible) is
1 if n is even x[n]=(−1)n =ejπn = −1 ifnisodd.
Units of frequency in discrete time
Most of the time in this unit we describe discrete frequency in terms of radians/sample. This unit is most convenient mathematically. It is also very useful to consider the more intuitive unit of cycles/sample. If ω0 is the frequency in radians/sample and f0 is the frequency in cycles/sample, then these are related by
ω0 = 2πf0 = 2π. T0
If we take the sum of two DT unit complex exponentials, the sum may or may not be periodic, because the DT unit complex exponentials individually may or may not be periodic. However, if we take the sum of two periodic DT complex exponentials, the result is always periodic.
Periodicity of sum of periodic DT complex exponentials
Let x[n] = αx1[n] + βx2[n] where x1[n] = ejω1n and x2[n] = ejω2n (and α, β ̸= 0). Assume that x1 is periodic with fundamental period T1 and x2 is periodic with fundamental period T2. Then x is periodic with fundamental period T0 given by the least common multiple of
T1 and T2.
Summary of Section 5.1
Unit complex exponentials are the basic building blocks of frequency-domain represen- tations of signals. If CT, any unit complex exponential is periodic, but the sum of two unit complex exponentials at different frequencies is only periodic if the frequencies are harmonically related. Not all DT unit complex exponentials are periodic, but the sum of two periodic DT unit complex exponentials is again periodic. There is a smallest period (two samples) and a largest frequency (π rad/sample) in discrete-time.
5.2 Finite duration signals
We say that a CT signal x is finite (or has finite duration) if there is an interval [a,b] for which x(t) = 0 when t < a and x(t) = 0 when t > b. Intuitively, a finite signal ‘starts’ at time a and ‘ends’ at time b. We say that it has duration b − a.
Similarly, a DT signal x is finite (of has finite duration) if there are discrete times A ≤ B for whichx[n]=0whenn≤A−1andx[n]=0whenn≥B+1. Again,wecanthinkofafinite signal as ‘starting’ at time n = A and ‘ending’ at time N = B and having duration B − A + 1.
5.2.1 Periodic extension of a finite duration signal
If x is a finite signal, we can build a periodic signal out of it by copying and pasting shifted copies of x. This makes a signal that is periodic with period equal to the duration of the original signal.
CT periodic extension
Suppose x is a finite duration CT signal such that x(t) = 0 unless t ∈ [a, b]. Then we can construct a new signal xperiodic with period T = b − a by
xperiodic(t) = x(t − mT ). m=−∞
This is called a shift-and-add summation and is illustrated below: x(t)
xperiodic(t)
a−2T a−T a a+T
DT periodic extension
Suppose x is a finite duration DT signal such that x[n] = 0 unless A ≤ n ≤ B. Then we can construct a new signal xperiodic with period N = B − A + 1 by
xperiodic[n] = x[n − mN]. m=−∞
This is called a shift-and-add summation and is illustrated below: x[n]
t A−2N A−N A A+N
AB xperiodic[n]
In certain cases, the fundamental period of xperiodic may be different from the duration of x. For example, in CT, this could happen if the original finite duration signal x looks like
since in this case the fundamental period of xperiodic would be T/2, even though the duration
of x is T. In DT there is the additional complication that if the finite duration signal x has 6
duration one (i.e., it is a shifted, scaled, unit impulse), then the periodic extension is constant! Usually, though, the duration of x and the fundamental period will be the same.
5.3 Continuous-time complex Fourier series
We have seen that it can be useful to express a periodic signal as a sum of sinusoids with different frequencies. In this section we discuss how to systematically find decompositions of periodic signals as sums of unit complex exponentials at different frequencies. This representation of a periodic signal is often called its complex Fourier series, and is introduced in Section 5.3.1. You may have seen Fourier series before using sines and cosines. The complex version, although initially more abstract, is easier to work with and generalises better. We explain the relationship between complex and trigonometric Fourier series in Section 5.3.5. As we saw in Section 5.2, there is a natural way to relate finite signals and periodic signals. In Section 5.3.2 we use this to develop complex Fourier series expressions for finite signals.
5.3.1 Continuous-time complex Fourier series
Summary of Section 5.2
A signal has finite duration if it is only non-zero over a finite time-period. We can make a periodic signal, called a periodic extension out of any finite duration signal.
If x is a CT complex periodic signal with fundamental period T0 = 2π then ω0
Xk = T x(t)e
x(t) = k=−∞
The following points are important to keep in mind when using and interpreting (5.4) and (5.5):
• The integral in (5.5) is written with limits 0 and T0. It is just as good to integrate over any single period [T, T + T0] of x. This is because for any integer k, the integrand x(t)ejkω0t is periodic, with period T0. This means that for any T,
1 T0 jkω0t 1 T+T0 jkω0t
Xk = T x(t)e dt= T x(t)e dt.
This additional flexibility can be very helpful when computing Fourier series coefficients.
• The equality in (5.4) should be interpreted with some caution since the infinite sum on the right-hand side may not converge uniformly. Sufficient conditions that ensure no convergence problems occur are briefly discussed in Section 5.3.4.
• The Fourier series decomposition of a signal is unique. In other words, if ∞∞
x(t) = Xkejkω0t = Zkejkω0t
then it must be the case that Xk = Zk for all integers k. This means that if a signal is written as a sum of complex exponentials at different frequencies, then we can simply read off the Fourier coefficients (see Example 5.2).
Example 5.1: Fourier series of periodic square wave
Let x(t) be the signal with period T0 that is defined for −T0/2 ≤ t ≤ T0/2 by
This signal is shown below
ifT1 <|t|
x(t)= Xkejkω0t fora≤t≤b k=−∞
where ω0 = 2π/(b − a) and
1 b −jkω0t Xk = T x(t)e
The idea is to take the periodic extension xperiodic of x with fundamental period T0. This is now a periodic signal, and so has a Fourier series decomposition
that is valid for (almost all) t.1 foralla≤t≤b. Sowehave
xperiodic(t) = Xkejkω0t k=−∞
By the definition of the periodic extension, xperiodic(t) = x(t)
x(t)= Xkejkω0t fora≤t≤b k=−∞
which gives a Fourier series decomposition of the finite duration signal x that is valid only for a ≤ t ≤ b. Furthermore, since a ≤ t ≤ b describes one period of xperiodic,
1b −jkω0t 1b −jkω0t Xk = T xperiodic(t)e dt = T x(t)e
0a 0a 1Under the Dirichlet assumptions on x, for instance.
5.3.3 Properties of CT Fourier series
One very useful way to find the Fourier series decomposition of a periodic signal y, is to relate y to a different signal x for which the Fourier series decomposition is already known. Very often the Fourier series coefficients of y can then be directly related to the Fourier series coefficients of x.
In this section we summarize some of these relationships. We mostly focus on situations in which the relationship between y and x is not described by an LTI system. This is because the next topic is devoted to understanding the relationship between the Fourier series coefficients of y and x when x is the input, and y is the output, of an LTI system.
Assume that x and z are both periodic with fundamental period T0 = 2π/ω0, and that the kth Fourier series coefficient of x is Xk, and the kth Fourier series coefficient of z is Zk.
If α and β are complex numbers then y = αx + βz is periodic with fundamental period T0. The Fourier series coefficients of y are
Yk = αXk + βZk for all k.
Modulation
If M is an integer then y(t) = ejMω0tx(t) is periodic with fundamental period T0. The Fourier series coefficients of y are
Yk = Xk−M for all integers k.
In other words, multiplication by a complex exponential at an integer multiple of the fundamental frequency causes a shift in the Fourier series coefficients.
To see why this holds, we replace x with its Fourier series expansion and note that ∞∞
y(t) = ejMω0t Xlejlω0t = Xlej(l+M)ω0t. l=−∞ l=−∞
Making the change of variables k = l + M in the sum we obtain
y(t) = Xk−M ejkω0t
k=−∞ from which it follows that Yk = Xk−M for all k.
Periodic convolution
If x and z are periodic with fundamental period T0 and T0
x(τ)z(t − τ) dτ
(where the integral can be taken over any single period of x and z) then y is periodic with
fundamental period T0. The Fourier series coefficients of y are Yk = T0XkZk for all integers k.
To see why this holds, we replace x(τ) and z(t − τ) in the integral with their Fourier series expansions to obtain
k=−∞ l=−∞ The integral can be simplified to
Xkejkω0τ Zlejlω0(t−τ) dτ.
Exchanging the order of summation and intergation we obtain
∞∞ T0 XkZlejlω0t
ej(k−l)ω0τ dτ.
if k = l otherwise.
ej(k−l)ω0τ dτ = 0
Substituting this into (5.6) and observing that only terms in the sum with k = l remain, gives ∞
y(t) = (T0XkZk)ejkω0t. k=−∞
Multipication
If y(t) = x(t)z(t) then y is periodic with fundamental period T0. The Fourier series
coefficients of y are
Yk = XlZk−l for all integers k,
the convolution of the Fourier series coefficients of x and z.
To see why this holds, we replace x and z with their Fourier series expansions to obtain ∞∞∞∞
y(t) = Xlejlω0t Zl′ ejl′ω0t = XlZl′ ej(l+l′)ω0t. l=−∞ l′=−∞ l=−∞ l′=−∞
We then use the formula for Yk, and exchange the order of integration and summation T0 ∞ ∞
Yk = T1 XlZl′ ej(l+l′)ω0te−jkω0t dt 0 0 l=−∞l′=−∞
= XlZl′ T ej(l+l −k)ω0t dt.
l=−∞l′=−∞ 0 0 To simplify we can use the fact that
1 j(l+l′−k)ω0t 1 ifl+l =k
T e dt = 0 otherwise. 00
This means that only the terms of the sum wi
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