−0.22 0 0.13 1
sinc(t) (light) and sinc2(t) (dark)
Area of any sinc or sinc2 equals area of triangle ABC inscribed within the main lobe.
B C A Time t [seconds] Amplitude sinc sinc squared
Figure 2.1 The sinc function and the sinc-squared function
REVIEW PANEL 2.18
The Signal sinc(t) Has Even Symmetry, Unit Area, and Unit-Spaced Zero Crossings sinc(t) =sin(πt)
πt sinc(0) = 1 sinc(t) = 0 when t = ±1, ±2, . . .
The sinc function is square integrable (an energy signal) but not absolutely integrable.
2.6
The Impulse Function
Loosely speaking, an impulse is a tall, narrow spike with finite area. An informal definition of the unit impulse function, denoted by δ(t) and also called a delta function or Dirac function, is
δ(t) = - 0, t ̸= 0 ∞, t = 0 ! ∞ −∞ δ(τ ) dτ = 1 (2.18)
It says that δ(t) is of zero duration but possesses finite area. To put the best face on this, we introduce a third, equally bizarre criterion that says δ(t) is unbounded (infinite or undefined) at t = 0 (all of which would make any mathematician wince).
REVIEW PANEL 2.19
An Impulse Is a Tall Narrow Spike with Finite Area and Infinite Energy ! ∞ −∞ δ(τ ) dτ = 1 δ(t) = - 0, t ̸= 0 ∞, t = 0
2.6.1
Impulses as Limiting Representations
A rigorous discussion of impulses involves distribution theory and the concept of generalized functions, which are beyond our scope. What emerges is that many sequences of ordinary functions behave like an impulse in their limiting form, in the sense of possessing the same properties as the impulse. As an example, consider the rectangular pulse (1/τ)rect(t/τ) of width τ and height 1/τ shown in Figure 2.2.
22 Chapter 2 Analog Signals
δ( t)
Width = τ Width = τ Width = τ
Area = 1 Area = 1 Area = 1 Area = 1 (1) t t 1/τ 1/τ 1/τ t t
Figure 2.2 The genesis of the impulse function
As we decrease τ, its width shrinks and the height increases proportionately to maintain unit area. As τ→ 0, we get a tall, narrow spike with unit area that satisfies all criteria associated with an impulse.
Signals such as the triangular pulse 1
τtri(t/τ), the exponentials τ1exp(−t/τ)u(t) and τ2exp(−|t|/τ), the sinc functions 1
τsinc(t/τ) and τ1sinc2(t/τ), the Gaussian τ1exp[−π(t/τ)2], and the Lorentzian τ/[π(τ2+ t2)] all possess unit area, and all are equivalent to the unit impulse δ(t) as τ → 0.
The signal δ(t − t0) describes an impulse located at t = t0. Its area may be evaluated using any lower and upper limits (say τ1and τ2) that enclose its time of occurrence, t0:
! τ2 τ1 δ(τ− t0) dτ = - 1, τ1< t0< τ2 0, otherwise (2.19)
Notation: The area of the impulse Aδ(t) equals A and is also called its strength. The function Aδ(t) is shown as an arrow with its area A labeled next to the tip. For visual appeal, we make its height proportional to A. Remember, however, that its “height” at t = 0 is infinite or undefined. An impulse with negative area is shown as an arrow directed downward.
2.6.2
Relating the Impulse and Step Function
If we consider the running integral of the unit impulse δ(t), it will equal zero until we enclose the impulse at t = 0 and will equal the area of δ(t) (i.e. unity) thereafter. In other words, the running integral of δ(t) describes the unit step u(t). The unit impulse may then be regarded as the derivative of u(t).
REVIEW PANEL 2.20 The Impulse Function Is the Derivative of the Step Function
δ(t) = d u(t)
dt u(t) =
! t −∞
δ(t) dt
2.6.3
Derivative of Signals with Jumps
Since a signal with sudden jumps may be described by step functions, the derivative of such signals must give rise to impulses. For example, the derivative of x(t) = Au(t)−Bu(t−α) is given by x′(t) = Aδ(t)−Bδ(t−α). This describes two impulses whose strengths A and −B correspond to the jumps (up and down) at t = 0 and t = α. In general, the derivative at a jump results in an impulse whose strength equals the jump. This leads to a simple rule for finding the generalized derivative of an arbitrary signal x(t). For its piecewise continuous portions, we find the ordinary derivative x′(t), and at each jump location in x(t), we include impulses whose strength equals the jump.
2.6 The Impulse Function 23 REVIEW PANEL 2.21
The Derivative of a Signal with Jumps Includes Impulses at the Jump Locations To find the derivative: Sketch the signal and check for any jumps.
At each jump location, include impulses (of strength equal to the jump) in the ordinary derivative.
EXAMPLE 2.6 (Derivative of Signals with Jumps) (a) Sketch the derivative of y(t) = rect(t/2).
Since y(t) = rect(t/2) = u(t + 1) − u(t − 1), its derivative is y′(t) = δ(t + 1) − δ(t − 1). The signals y(t) and y′(t) are sketched in Figure E2.6.
(t) h e -t 2 (t) y e -t 2 t −2 − 1 y(t) −1 1 t −1 1 t h(t) t 2 (2) (1) (−1)
Figure E2.6 The signals for Example 2.6
(b) Find the derivative of h(t) = 2e−tu(t).
The derivative is best found by sketching h(t) as shown. Its ordinary derivative is −2e−t. We also include an impulse of strength 2 at the jump location t = 0. Thus, h′(t) = 2δ(t) − 2e−tu(t). The signals h(t) and h′(t) are also sketched in Figure E2.6.
It is not such a good idea to use the chain rule and get h′(t) = 2e−tδ(t)− 2e−tu(t) because we must then use the product property (see the next section) to simplify the first term as 2e−tδ(t) = 2δ(t).
2.6.4
Three Properties of the Impulse
Three useful properties of the impulse function are the scaling property, the product property, and the sifting property.
The Scaling Property
An impulse is a tall, narrow spike, and time scaling changes its area. Since δ(t) has unit area, the time- compressed impulse δ(αt) should have an area of 1
|α|. Since the impulse δ(αt) still occurs at t = 0, it may be regarded as an unscaled impulse 1
|α|δ(t). In other words, δ(αt) = |α|1 δ(t). A formal proof is based on a change of variable in the defining integral. Since a time shift does not affect areas, we have the more general result: δ[α(t− β)] = 1 1 1 1α1 1 1 1 1 δ(t − β) (2.20)
24 Chapter 2 Analog Signals The Product Property
This property says that if you multiply a signal x(t) by the impulse δ(t − α), you get an impulse whose strength corresponds to the signal value at t = α. The product of a signal x(t) with a tall, narrow pulse of unit area centered at t = α is also a tall, narrow pulse whose height is scaled by the value of x(t) at t = α or x(α). This describes an impulse whose area is x(α) and leads to the product property:
x(t)δ(t− α) = x(α)δ(t − α) x(t)δ(t) = x(0)δ(t) (2.21)
The Sifting Property
The product property immediately suggests that the area of the product x(t)δ(t − α) = x(α)δ(t − α) equals x(α). In other words, δ(t− α) sifts out the value of x(t) at the impulse location t = α, and thus,
! ∞ −∞ x(t)δ(t− α) dt = x(α) ! ∞ −∞ x(t)δ(t) dt = x(0) (2.22)
This extremely important result is called the sifting property. It is the sifting action of an impulse (what it does) that purists actually regard as a formal definition of the impulse.
REVIEW PANEL 2.22 The Scaling, Product, and Sifting Properties of an Impulse
Scaling Product Sifting
δ(αt) = 1
|α|δ(t) x(t)δ(t− α) = x(α)δ(t − α)
! ∞ −∞
x(t)δ(t− α) dt = x(α)
EXAMPLE 2.7 (Properties of the Impulse Function)
(a) Consider the signal x(t) = 2r(t)−2r(t−2)−4u(t−3). Sketch x(t), f(t) = x(t)δ(t−1), and g(t) = x′(t). Also evaluate I ="∞
−∞x(t)δ(t− 2) dt. Refer to Figure E2.7A for the sketches.
x (t) t 2 3 4x(t) t 2 3 2 t x(t) δ 1 (2) (−4) (t −1)
Figure E2.7A The signals for Example 2.7(a)
From the product property, f(t) = x(t)δ(t − 1) = x(1)δ(t − 1). This is an impulse function with strength x(1) = 2.
The derivative g(t) = x′(t) includes the ordinary derivative (slopes) of x(t) and an impulse function of strength −4 at t = 3.
By the sifting property, I ="∞
−∞x(t)δ(t− 2) dt = x(2) = 4.
(b) Evaluate z(t) = 4t2δ(2t− 4).
2.6 The Impulse Function 25 Thus, z(t) is an impulse of strength 8 located at t = 2, a function we can sketch.
(c) Evaluate I1="0∞4t2δ(t + 1) dt.
The result is I1= 0 because δ(t + 1) (an impulse at t = −1) lies outside the limits of integration. (d) Evaluate I2="−42 cos(2πt)δ(2t + 1) dt.
Using the scaling and sifting properties of the impulse, we get I2=
! 2
−4cos(2πt)[0.5δ(t + 0.5)] dt = 0.5 cos(2πt)|t=−0.5= −0.5
2.6.5
Signal Approximation by Impulses
A signal x(t) multiplied by a periodic unit impulse train with period tsyields the ideally sampled signal, xI(t), as shown in Figure 2.3. The ideally sampled signal is an impulse train described by
xI(t) = x(t) ∞ 2 k=−∞ δ(t− kts) = ∞ 2 k=−∞ x(kts)δ(t − kts) (2.23) s t xI(t) s t t
Ideally sampled signal
x(t)
t
Analog signal i(t) Multiplier
t
(1) (1)
Sampling function
Figure 2.3 The ideally sampled signal is a (nonperiodic) impulse train
Note that even though xI(t) is an impulse train, it is not periodic. The strength of each impulse equals the signal value x(kts). This form actually provides a link between analog and digital signals.
REVIEW PANEL 2.23
Ideally Sampled Signal: An Impulse Train Whose Strengths Are the Signal Values xI(t) =
∞ 2 k=−∞
x(kts)δ(t − kts)
To approximate a smooth signal x(t) by impulses, we section it into narrow rectangular strips of width ts as shown in Figure 2.4 and replace each strip at the location kts by an impulse tsx(kts)δ(t − kts) whose strength equals the area tsx(kts) of the strip. This yields the impulse approximation
x(t)≈ ∞ 2 k=−∞
26 Chapter 2 Analog Signals
s
t ts
Section signal into narrow rectangular strips
t
Replace each strip by an impulse
t
Figure 2.4 Signal approximation by impulses
Naturally, this approximation improves as ts decreases. As ts→ dλ → 0, kts takes on a continuum of values λ, and the summation approaches an integral
x(t) = lim ts→0 ∞ 2 k=−∞ tsx(kts)δ(t − kts) = ! ∞ −∞ x(λ)δ(t− λ) dλ (2.25)
A signal x(t) may thus be regarded as a weighted, infinite sum of shifted impulses. REVIEW PANEL 2.24
We Can Approximate a Signal by a Sum of Impulses (Tall, Narrow Pulses) The area of the kth impulse is the pulse width tstimes the signal height x(kts).
x(t) ≈ ts ∞ 2 k=−∞ x(kts)δ(t − kts)
2.7
The Doublet
The derivative of an impulse is called a doublet and denoted by δ′(t). To see what it represents, consider the triangular pulse x(t) = 1
τtri
+t τ ,
of Figure 2.5. As τ → 0, x(t) approaches δ(t). Its derivative x′(t) should then correspond to δ′(t). Now, x′(t) is odd and shows two pulses of height 1/τ2 and −1/τ2 with zero area. As τ → 0, x′(t) approaches +∞ and −∞ from below and above, respectively. Thus, δ′(t) is an odd function characterized by zero width, zero area, and amplitudes of +∞ and −∞ at t = 0. Formally, we write
δ′(t) = - 0, t̸= 0 undefined, t = 0 ! ∞ −∞ δ′(t) dt = 0 δ′(−t) = −δ′(t) (2.26) The two infinite spikes in δ′(t) are not impulses (their area is not constant), nor do they cancel. In fact, δ′(t) is indeterminate at t = 0. The signal δ′(t) is therefore sketched as a set of two spikes, which leads to the name doublet. Even though its area"δ′(t) dt is zero, its absolute area" |δ′(t)| dt is infinite.
2.7.1
Three Properties of the Doublet
The doublet also possesses three useful properties. The scaling property comes about if we write the form for the scaled impulse
δ[α(t− β)] = 1
|α|δ(t− β) (2.27)
and take the derivative of both sides to obtain αδ′[α(t − β)] = 1
|α|δ
′(t − β) δ′[α(t − β)] = 1 α|α|δ