I'm having trouble finding this sum

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In summary, the sum of e^{-j\omega n} from -N1 to +N1 can be simplified using the formula \sum_{k=0}^{N}x^{k} = \frac{1-x^{N+1}}{1-x}. By letting \lambda = e^{-j\omega}, we can express the sum as \frac{e^{j\omega N1}-e^{-j\omega N1}e^{-j\omega}}{1-e^{-j\omega}}. Multiplying the numerator and denominator by e^{j\omega/2} and using the identity e^{jx}-e^{-jx}=2jsin(x), we can simplify the expression to \frac
  • #1
Jncik
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Homework Statement



find

[tex]\sum_{-N1}^{+N1}e^{-j\omega n}[/tex]

Homework Equations





The Attempt at a Solution



Let [tex]\lambda = e^{-j\omega} [/tex]

we have

[tex]\sum_{-N1}^{+N1}\lambda ^{n} = \sum_{-N1}^{-1}\lambda ^{n} + \sum_{0}^{+N1} \lambda ^{n}[/tex]

for the first i have

[tex]S = \lambda ^{-N1} + \lambda ^{-N1+1} + \lambda ^{-N1+2} + ... + \lambda ^{-2} + \lambda ^{-1}[/tex]

[tex]-\lambda S = -\lambda ^{-N1+1} - \lambda ^{-N1+2} - \lambda ^{-N1+3} - ... - \lambda ^{-1} - \lambda ^{0} [/tex]

hence

[tex] S = \frac{\lambda ^{-N1} - 1}{1-\lambda }[/tex]

for the second i have

[tex] S2 = \lambda^{0} + \lambda^{1} + ... + \lambda^{N1-1} + \lambda^{N1}[/tex]
[tex] -\lambda S2 = -\lambda^{1} - \lambda^{2} - ... - \lambda^{N1} - \lambda^{N1+1}[/tex]

hence

[tex] S2 = \frac{1 - \lambda^{N1+1}}{1-\lambda}[/tex]

so the sum is

[tex]\frac{1-\lambda^{N1+1} + \lambda^{-N1} - 1}{1-\lambda} = \frac{\lambda^{-N1} - \lambda^{N1+1}}{1-\lambda} = \frac{e^{j \omega N1} - e^{-j \omega N1}e^{-j\omega}}{1-e^{-j\omega}}[/tex]

but the book says [tex]\frac{sin\omega(N1 + \frac{1}{2})}{sin(\frac{\omega}{2})}[/tex]
 
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  • #2
Multiply numerator and denominator by e^(j*w/2). Now remember e^(jx)-e^(-jx)=2jsin(x).
 
  • #3
thanks a lot :)
 

FAQ: I'm having trouble finding this sum

Why am I having trouble finding this sum?

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Yes, there are certain rules and order of operations that should be followed when solving a sum. This includes solving parentheses or brackets first, then exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right. Remember to always double-check your calculations and follow the correct order of operations to ensure an accurate answer.

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