Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 274

0.5

|X(f)| |R(f)| A

f f

0 5 20 4 0 0 5 20 4 0

0.47 f s =18.8

18.8

B 1 B 2

|P(f)| |V(f)|

f f , Hz

0 5 20 35 40 0 5 20 35 40

|Y(f)| C

D

f

0 5 20 35 40

Tutorial 39

Q: If the interest rate r is fixed and there are no account fees, then a savings
account in a bank represents an LTI system. Let x(n) denote the amount of money
deposited (or withdrawn) in the nth day, and y(n) be the total amount of money in
the account at the end of the nth day. Assume r is compounded daily.

1. Find the impulse response h(n) of the system using a time-domain approach.
2. Find the system output [using h(n)] at the nth day.
3. Find the system difference equation, the transfer function H(z), and the impulse
response h(n). Can we use the fft to find the output y(n)?

Solution:
1. We need an input which is a delta function d(n). Put $1 in the opening day
(n = 0) and $0 afterwards. Now we have:

y ð 0 Þ¼ x ð 0 Þ¼ d ð 0 Þ¼ 1; y ð 1 Þ¼ 1 þ r;
y ð 2 Þ¼ y ð 1 Þþ ry ð 1 Þ¼ y ð 1 Þð 1 þ r Þ¼ð 1 þ r Þ 2 : Similarly: y ð n Þ¼ð 1 þ r Þ n :

This is the impulse response of the system, {h(n)}.
2. y ð n Þ¼ x ð n Þ h ð n Þ¼ P k ¼ 0 x ð k Þ h ð n k Þ¼ P k ¼ 0 x ð k Þ½ 1 þ r n k :
For example, if r = 0.01% = 0.0001 and we put $100 in the opening day, $50
in the next day, $50 in the 3rd day, and nothing afterwards, then at the end of
the year we have: y ð 365 Þ¼ x ð 0 Þð 1 þ r Þ 365 þ x ð 1 Þð 1 þ r Þ 364 þ x ð 1 Þð 1 þ r Þ 363 ¼
$207 : 41 :
3. New balance = old + r old + new deposit