18.034 - Honors Differential Equations, Spring 2007                                           Dr. Vera Mikyoung Hur


Convolution. Consider a linear system in which the effect at the present time t of a stimulus
f (t1 )dt1 at the past time t1 is proportional to the stimulus. On physical grounds we assume that
the proportionality constant depends only on the elapsed time t − t1 and hence is of the form
g(t − t1 ). The effect at the present time t is therefore
                                                       f (t1 )g(t − t1 )dt1 .
Since the system is linear, the response to the history can be obtained by adding these effects and
we are lead to the integral
                                            � t
(1)                                  w(t) =     f (t1 )g(t − t1 )dt1 .
(The lower limit 0 means that the process is assumed to have started at time t = 0.)
   The expression (1) is called the convolution of f and g. It gives the response at the present time
t as a weighted superposition over the inputs at times t1 � t. The weighting factor g(t − t1 )
characterizes the system and f (t1 ) characterizes the history of the input.
   The function w in (1) is denoted by f ∗ g.
   The following theorem show that convolution in the t-domain corresponds to multiplication in
the s-domain.
Theorem 1. If f, g ∈ E then f ∗ g ∈ E and L[f ∗ g] = L[f ]L[g].
Proof. I leave the proof of f ∗ g ∈ E to the reader. By defining f (t) = 0 and g(t) = 0 for all negative
t, we write
                                        � ∞ �� ∞                        �
                          L[f ∗ g(t)] =             f (t1 )g(t − t1 )dt1 e−st dt
                                          −∞    −∞
                                        � ∞ �� ∞                      �
                                      =             g(t − t1 )e dt f (t1 )dt1 .
                                                      −∞       −∞

We set t − t1 = t2 , t = t1 + t + 2, and the assertion follows.                                                  �

   The convolution plays a prominent role in the study of heat conduction, wave motion, plastic
flow and creep, and in other branches of mathematical physics. It is also encountered in sociology,
economics, ecology, and genetics in which an effect at a given time t1 induces a delayed response
at a later time t.
Example 2. Consider the problem
                                       y �� + ω 2 y = ω 2 f (t),        y(0) = y � (0) = 0,
where ω is a constant and f ∈ E. Taking the Laplace transform yields that
                                                      Ly =               L[f ],
                                                               s2   + ω2
                                                L[y] = L[f ]L[g] = L[f ∗ g],
where g(t) = ω sin ωt. By uniqueness, then y = f ∗ g, or more explicitly
                                         � t
                                y(t) = ω     f (t1 ) sin ω(t − t1 )dt1 .

The unit impulse function. We consider the response of a system to a narrow spike function
(an impulse) that acts for a very short time but produces a large effect, e.g., a point charge or an
electron point. To formulate the idea of an impulse, let a be a small positive constant and let δa (t)
be the function defined as
                                                  for 0 � t < a,
(2)                              δa (t) = a
                                           0      elsewhere.

Note that the area under the curve is the unity while the function is nonzero only on a small
interval. Its Laplace transform is
                                                          1 − e−sa
                                             � a
                                                 1 −st
                                 L[δa (t)] =       e dt =          .
                                              0− a           sa
It is a conceptual aid to introduce an expression δ(t) that describes the effect of δa (t) as a → 0
and to say that L[δ(t)] = 1. The symbol δ(t) is called the unit impulse function or the Dirac delta
distribution. is loosely thought of as a function on the real line which is zero everywhere except
at the origin,
                                               δ(t) = 0             for x �= 0,
which is subject to
                                                     �    ∞
                                                              δ(t)dt = 1.
  Let me say at the outset that δ(t) is not truly a function. Rigorous treatment of the Dirac delta
requires the theory of distributions∗.
Example 3 (The unit impulse response). We consider the following RLC- circuit problem with the
rest initial conditions
                                 (D2 + 2D + 2)x = f (t),                 x(0) = x� (0) = 0.
The unit impulse response is the solution x(t) to a unit impulse function δ(t), that is, the solution to
                                  (D2 + 2D + 2)x = δ(t),                x(0) = x� (0) = 0.
Taking the Laplace transform and using that L[δ(t)] = 1, we obtain in succession
                                  L[x] =                 ,            x(t) = u(t)e−t sin t.
                                            (s + 1)2 + 1
It is immediate to see that the unit impulse input has a lasting effect.
    We note that the solution x is continuous for all t and it satisfies the differential equation for
t = 0. At t = 0, however, it satisfies neither the differential equation nor the initial conditions, and
it is not even differentiable. Indeed, x� (0−) = 0 and x� (0+) = 1. The unit impulse thus produces a
jump of magnitude 1 in x� (t) at t = 0.

    A distribution is characterized, not by giving its value δ(t) at each t, but by giving its value δφ on a suitable class of
functions φ. Distributions were introduced in mid 1930s by Sergei Sobolev, and independently, in late 1940s by Laurent
Test functions. The basis of the theory of δ(t) rests on the fact that δ(t) is not a function whose
value is defined for each t, but is defined via the action of δ(t) on other functions φ(t), which
is called test functions. In the general theory of distributions, test functions are assumed to have
derivatives of all orders and have support on a finite interval. Here we require that φ(t) is contin­
uously differentiable as many times as are needed.
   The defining characteristic of the Dirac delta is
                                        � ∞
(3)                                          δ(t)φ(t)dt = φ(0).

If δ(t) were an ordinary function and if the integral in (3) were a ordinary Riemann integral, then
a change of variable would give
                             � ∞                   � ∞
                                  δ(t − c)φ(t)dt =       δ(t)φ(t + c)dt.
                                −∞                       −∞

By definition in (3), the right side is φ(c). This is now taken as the definition of the left side, that is,
                                       � ∞
                                            δ(t − c)φ(t)dt = φ(c).

Similarly, we defined the derivative of δ(t) by
                               � ∞                �      ∞
                                    δ (t)φ(t)dt =             δ(t)φ� (t)dt.
                                     −∞                  −∞

Approximate identity. The delta function can be viewed as the limit of a sequence of functions
δa (t) in (2). The term approximate identity has a particular meaning in harmonic analysis, in relation
to a limiting sequence to an identity element for the convolution operation. Another kind of
nascent delta functions is the normal distribution
                                                     1      2 2
                                           δa (t) = √ e−x /a .
                                                   a π

The transfer function. Let p(D) be a linear differential operator with constant coefficient and with
the characteristic polynomial p(s). The DE
                                                p(D)x = Aest
could be solved, in general, by trying the solution x = Best . Under the assumption that p(s) �= 0,
the result is p(s)Best = Aest , or B = p(s) A. The function W (s) = 1/P (s) that transforms the input
amplitude A into the output amplitude B is called the transfer function. The major property of a
transfer function is given by
                                 (output)=(transfer function)(input).
  Suppose now that we characterize the input f and output x by their Laplace transforms F and
X. If f ∈ E the rest solution of the equation p(D)x = f satisfies
                                   p(s)Lx = Lf       or X = W (s)F,
where W (s) = 1/p(s) is the transfer function as defined above. This allows an arbitrary input f .
Note that the passage from F to X involves merely a multiplication, while the passage from f to
x requires the solution of a differential equation. This is one of the advantages of working in the
The problem of identification. Here we put in a known input, measure the output, and try to
find the differential equation. This is a common view in bio-mathematics and in mathematical
medicine. After measuring the response to a known regime of drug dosage, for example, one can
try to discover the mechanism by which that response was produced.
   It should be said at the outset that the most one can hope for is to find the coefficients in the
operator p(D), and if many choices of system parameters give the same p(D) then the situation
cannot be further disentangled by mere study of the input-output relation. As an illustration, the
vibrations of a frictionless mass-spring system are characterized by two parameters m and k, the
mass and the stiffness constant. However, even complete knowledge of the differential equation
                                          mx�� + kx = kf (t)
yields only the ratio k/m, not k and m separately.
   The differential operator p(D) is determined by its characteristic polynomial p(s) and succes­
sively p(s) is determined by the transfer function W (s) = 1/p(s). The input function f (t) ∈ E
is said to be nontrivial if its transform F (s) is not identically zero. The output associated with
any nontrivial inpput suffices for the complete determination of the transfer function W (s). For
example, the inputs f1 (t) = u(t) and f2 (t) = δ(t) yields, respectively,
                              W (s) = sX1 (s) and W (s) = X2 (s).
The solutions corresponding to X1 and X2 are called, respectively, the unit step response and the
unit impulse response. Note that the second equation states that the transfer function is the unit
impulse response.

Lumped-parameter circuits. The voltage V across a capacitor is related to the charge Q by V =
Q/C where 1/C is the elastance. Since the current is I = dQ/dt an integration gives
                                              � t
                                V = Q/C = S       Idt + Q(0)/C.
The initial charge Q(0) is typically 0 and this condition is assumed here.
  If the capacitor is connected in series with a resistor of resistance R, an inductor of inductance
L and a source of voltage V (t) satisfy
                                                   � t
                                    L + RI + S         Idt = V (t).
                                      dt            0
The transform of this with I(0) = 0 is
                                     (sL + R + s−1 S)L[I] = L[V ].
The conditions Q(0) = 0 and I(0) = 0 are used here.
  For a circuit with n independent loops a calculation of the same sort can be carried out for each
loop. The result after transformation is a system of the form
                                 (sLij + Rij + s−1 Sij )L[Ij ] = L[Vi ],

where Ij is the current and Vj is the impressed voltage associated with the jth loop. The resistences
Rij and elastances Sij are 0 for i �= j but the inductances Lij need not have this property because
of the possibility of mutual inductance between branches.
   In the matrix form, the above is written
                                         V = ZI,       I =YV
where Z = sL + R + s−1 S and Y = (sL + R + s−1 S)−1 . In the first case, I is considered to be input,
V is output, and the transfer function Z is an impedance. In the second case V is considered to be
input, I is output, and Y is an admittance.
   Circuits that are characterized by numbers Rij , Lij Sij as above are said to be lumped parameter,
to distinguish them from distributed parameter systems such as a waveguide or a submarine cable.

                                                   R EFERENCES
[1] Garret Birkhoff & Gian-Carlo Rota, Ordinary Differential Equations, 4th ed., New York: Wiley, 1978.
[2] Ray Redheffer & Dan Port Differential Equations, theory and applications, Jones and Barlett publishers, Boston, 1991.

1. If f is continuous at t = c, where c > 0, show that
                             lim u(t − c)f (t) = 0,            lim h(t − c)f (t) = f (c).
                            t→c−                               t→c+
Hence u(t − c)f (t) is continuous at c if and only if f (c) = 0. (Hint: Continuity at c means
limt→c f (t) = f (c). )
2. If x = 0 for t < 0 and x� (t) = δ(t) for t > 0 then the Laplace transform suggests that sLx = 1.
Assuming this, conclude that x agrees with the unit step function u(t) except perhaps at t = 0. In
that sense δ(t) = u� (t).


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