In the literature of digital signal processing, we often encounter geometric series expressions like
or
Equation B2
Unfortunately, many authors make a statement like "and we know that," and drop Eqs. (B1) or (B2) on the unsuspecting reader who's expected to accept these expressions on faith. Assuming that you don't have a Ph.D. in mathematics, you may wonder exactly what arithmetic sleight of hand allows us to arrive at Eqs. (B1) or (B2). To answer this question, let's consider a general expression for a geometric series such as
Equation B3
where n, N, and p are integers and a and r are any constants. Multiplying Eq. (B3) by r, gives us
Equation B4
Subtracting Eq. (B4) from Eq. (B3) gives the expression
or
Equation B5
So here's what we're after. The closed form of the series is
Equation B6
(By closed form, we mean taking an infinite series and converting it to a simpler mathematical form without the summation.) When a = 1, Eq. (B6) validates Eq. (B1). We can quickly verify Eq. (B6) with an example. Letting N = 5, p = 0, a = 2, and r = 3, for example, we can create the following list:
n 
arn = 2 · 3n 

0 
2 · 30 = 2 
1 
2 · 31 = 6 
2 
2 · 32 = 18 
3 
2 · 33 = 54 
4 
2 · 34 = 162 
The sum of this column is 
Plugging our example N, p, a, and r values into Eq. (B6),
Equation B7
which equals the sum of the rightmost column in the list above.
As a final step, the terms of our earlier Eq. (B2) are in the form of Eq. (B6) as p = 0, a = 1, and r = e–j2pm/N.[] So plugging those terms from Eq. (B2) into Eq. (B6) gives us
[] From the math identity axy = (ax)y, we can say e–j2pnm/N = (e–j2pm/N)n, so r = e–j2pm/N.
Equation B8
confirming Eq. (B2).
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