The convolution sum in Eq. (5.23) can be expressed in a matrix form as
y = Hx,
Notice that matrix H is a lower-triangular matrix (all elements above the main diagonal are zero). Knowing h[n] and the output y[n], we can determine the input x[n] as
x = H−1
y.
This operation is the reverse of the convolution and is known as deconvolution. Moreover, knowing x[n] and y[n], we can determine h[n]. This can be done by expressing the matrix equation as n + 1 simultaneous equations in terms of n + 1 unknowns h[0], h[1], ... , h[n]. These equations can readily be solved iteratively. Thus, we can synthesize a system that yields a certain output y[n] for a given input x[n].
(a) Design a system (i.e., determine h[n]) that will yield the output sequence (8, 12, 14, 15, 15.5, ...) for the input sequence (1, 1, 1, 1, 1, ...).
(b) For a system with the impulse response sequence (1, 2, 4, ...), the output sequence is (1, 7/3, 43/9, ...). Determine the input sequence.