Help on prooving a summation formula

In summary, the conversation is about proving a formula involving an arbitrary function and two summations. The first summation is from t=1 to N and the second summation is from s=1 to N, while the second equation involves a summation from τ=-N+1 to N-1. The person asking for help has tried small examples and is considering changing the variables in the first double summation, but is stuck. The other person suggests considering τ=t-s as a change of index and asks for clarification on the limits of the summations in that case.
  • #1
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Homework Statement


I need some advice on prooving this formula (f is an arbitrary function):

[itex]\sum^{N}_{t=1}[/itex][itex]\sum^{N}_{s=1}[/itex]f(t-s)=[itex]\sum^{N-1}_{τ=-Ν+1}[/itex](N-|τ|)f(τ)

Thanks in advance
 
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  • #2
What have you done so far? Have you tried small examples, like N = 2 or N = 3?

RGV
 
  • #3
Well its easy to see that it works with examples like N=2 or N=3. For example for N=2 the value of both sides is f(1)+f(-1)+2f(0). Same for N=3. I am thinking that maybe I should do a variables change in the first double sums, to end up to a more common summation formula, but I am kinda stuck.

EDIT: We can consider that s and t are integers, or that the arbitrary f() function represents a discrete time signal.
 
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  • #4
Looks to me like there is a change of index at work there. Since the left side of the equation involves f(t- s) and the other side f(τ), you should immediately think of τ= t- s.
 
  • #5
could you please help on how the limits of the left side sums would be in that case (τ=t-s)?
 

1. What is a summation formula?

A summation formula is a mathematical expression that allows you to calculate the sum of a series of numbers. It is denoted by a capital Greek letter sigma (Σ) and is followed by the numbers to be summed, along with their corresponding coefficients. For example, the summation formula for the series 1+2+3+4+5 would be written as Σn, where n represents the number being summed.

2. How do I prove a summation formula?

To prove a summation formula, you will need to use mathematical induction. This method involves showing that the formula holds true for a specific value (usually 1 or 0), and then proving that if the formula holds for a particular value, it also holds for the next value. By repeating this process, you can show that the formula holds true for all values in the series.

3. What are the steps to proving a summation formula using mathematical induction?

The steps to proving a summation formula using mathematical induction are as follows:

Step 1: Show that the formula holds true for the first value in the series (usually 1 or 0).

Step 2: Assume that the formula holds true for a particular value (usually denoted as k).

Step 3: Use this assumption to prove that the formula also holds true for the next value (k+1).

Step 4: Repeat this process for all values in the series, showing that the formula holds true for each value.

Step 5: Conclude that the formula holds true for all values in the series, thereby proving its validity.

4. Can a summation formula be used for infinite series?

Yes, a summation formula can be used for infinite series. However, in order for the formula to be valid, the series must be convergent, meaning that the sum of the series approaches a finite value as the number of terms increases. In these cases, the summation formula is denoted by the infinity symbol (∞), and the series is written as Σn=1 to ∞.

5. What are some common summation formulas?

Some common summation formulas include:

- Sum of integers: Σn = n(n+1)/2

- Sum of squares: Σn^2 = n(n+1)(2n+1)/6

- Sum of cubes: Σn^3 = (n(n+1)/2)^2

- Geometric series: Σar^n = a(1-r^n)/(1-r), where a is the first term and r is the common ratio

- Harmonic series: Σ1/n = ln(n+1), where ln is the natural logarithm

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