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$$\delta(n) - \frac{1}{2} \mbox{sinc} \left(\frac{n}{2}\right) = (-1)^n \frac{1}{2} \mbox{sinc} \left( \frac{n}{2} \right)$$

The picture shows what I've tried enter image description here

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  • $\begingroup$ I think your picture is wrong. $\delta(n)$ is only non-zero for $n=0$ $\endgroup$ – Hilmar May 5 '20 at 13:05
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Hint : Use the fact that, $\sin(-\frac{n\pi}{2}) = (-1)^n \sin(\frac{n\pi}{2}), \forall n \in \mathbb Z$

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  • $\begingroup$ I am still failing to prove it using your suggestion. Please check my picture. $\endgroup$ – M.Bore May 5 '20 at 11:39
  • $\begingroup$ In your picture , in the end you are getting right result. What is the problem? I could not understand. Can you please give a little more detail. $\endgroup$ – DSP Rookie May 5 '20 at 11:41
  • $\begingroup$ I am sceptical because the graphs don't quite show it $\endgroup$ – M.Bore May 5 '20 at 11:45
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    $\begingroup$ $ \delta(n) - \frac{1}{2} sinc (\frac{n}{2}) = (-1)^n \frac{1}{2} sinc( \frac{n}{2}) $ \\Given that $ \sin{\frac{-n\pi}{2}} = (-1)^n \sin{\frac{n\pi}{2}} $ \\ $ \delta(n) - \frac{\frac{1}{2}\sin{\frac{\pi n}{2}} }{\pi \frac{n}{2}} $ = $ \frac{\frac{1}{2}\sin{\frac{-\pi n}{2}}}{\pi \frac{n}{2}} $ = $ \frac{1}{2}(-1)^n \frac{\sin{\frac{\pi n}{2}}}{\frac{\pi n}{2}} $ \\ = $ \frac{1}{2}(-1)^n \sin{\frac{n}{2}} $ can you confirm if that is true $\endgroup$ – M.Bore May 5 '20 at 12:33
  • $\begingroup$ yes this is absolutely right. $\endgroup$ – DSP Rookie May 5 '20 at 12:39
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You wrote "show" rather than "prove". Here's how you "show" your equality using MATLAB:

n = -7: 1 : 7;

delta = zeros(1, length(n));

delta (8) = 1

Left = delta-sinc(n/2)/2;

Right = (-1).^n.*sinc(n/2)/2;

figure(1), clf, hold on

plot(n, Left, '-bo', 'markersize', 6)

plot(n, Right, ':rd', 'markersize', 9); grid on, zoom on

title('Blue squares = Left side, Red dots = Right side'); xlabel('n');
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  • $\begingroup$ true! I corrected my question. Thank you for showing it, I'm still struggling to prove it though. $\endgroup$ – M.Bore May 5 '20 at 11:00
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$ \delta(n) - \frac{1}{2} sinc (\frac{n}{2}) = (-1)^n \frac{1}{2} sinc( \frac{n}{2}) $

Given that

$ \sin{\frac{-n\pi}{2}} = (-1)^n \sin{\frac{n\pi}{2}} $

$ \delta(n) - \frac{\frac{1}{2}\sin{\frac{\pi n}{2}} }{\pi \frac{n}{2}} $ =

$ \frac{\frac{1}{2}\sin{\frac{-\pi n}{2}}}{\pi \frac{n}{2}} $

$ \frac{1}{2}(-1)^n \frac{\sin{\frac{\pi n}{2}}}{\frac{\pi n}{2}} $

= $ \frac{1}{2}(-1)^n \sin{\frac{n}{2}} $

QED

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Hint: split it into two sequences: one for $n$ even and one for $n$ odd

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