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Common DTFTs

 Time Domain x[n] Frequency Domain X(w) Notes $δ [ n ] δ [ n ]$ 1 $δ[n-M]δ[n-M]$ $e-jwMe-jwM$ integer M $∑m=-∞∞δ[n-Mm]∑m=-∞∞δ[n-Mm]$ $∑ m = - ∞ ∞ e - j w M m = 1 M ∑ k = - ∞ ∞ δ ( w 2 π - k M ) ∑ m = - ∞ ∞ e - j w M m = 1 M ∑ k = - ∞ ∞ δ ( w 2 π - k M )$ integer M $e-jane-jan$ $2 π δ ( w + a ) 2 π δ ( w + a )$ real number a $u[n]u[n]$ $1 1 - e - j w + ∑ k = - ∞ ∞ π δ ( w + 2 π k ) 1 1 - e - j w + ∑ k = - ∞ ∞ π δ ( w + 2 π k )$ $anu(n)anu(n)$ $1 1 - a e - j w 1 1 - a e - j w$ if $|a|<1|a|<1$ $cos(an)cos(an)$ $π [ δ ( w - a ) + δ ( w + a ) ] π [ δ ( w - a ) + δ ( w + a ) ]$ real number a $sinc[(a+n)]sinc[(a+n)]$ $e j a w e j a w$ real number a $W·sinc2(Wn)W·sinc2(Wn)$ $t r i ( w 2 π W ) t r i ( w 2 π W )$ real number W, $0 < W ≤ 0 . 5 0 < W ≤ 0 . 5$ $W·sinc[W(n+a)]W·sinc[W(n+a)]$ $r e c t ( w 2 π W ) · e j a w r e c t ( w 2 π W ) · e j a w$ real numbers W,a $0 < W ≤ 1 0 < W ≤ 1$ $rect[(n-M/2)M]rect[(n-M/2)M]$ $sin[w(M+1)/2]sin(w/2)e-jwM/2sin[w(M+1)/2]sin(w/2)e-jwM/2$ integer M $W ( n + a ) { c o s [ π W ( n + a ) ] - s i n c [ W ( n + a ) ] } W ( n + a ) { c o s [ π W ( n + a ) ] - s i n c [ W ( n + a ) ] }$ $j w · r e c t ( w π W ) e j a w j w · r e c t ( w π W ) e j a w$ real numbers W,a $0 < W ≤ 1 0 < W ≤ 1$ $1 π n 2 [ ( - 1 ) n - 1 ] 1 π n 2 [ ( - 1 ) n - 1 ]$ $| w | | w |$ $0 n = 0 ( - 1 ) n n elsewhere 0 n = 0 ( - 1 ) n n elsewhere$ $j w j w$ differentiator filter $0 n odd 2 π n n even 0 n odd 2 π n n even$ $j w < 0 0 w = 0 - j w > 0 j w < 0 0 w = 0 - j w > 0$ Hilbert Transform
Notes

rect(t) is the rectangle function for arbitrary real-valued $tt$.

$rect(t) = 0 if | t | > 1 / 2 1 / 2 if | t | = 1 / 2 1 if | t | < 1 / 2 rect(t) = 0 if | t | > 1 / 2 1 / 2 if | t | = 1 / 2 1 if | t | < 1 / 2$
1

tri(t) is the triangle function for arbitrary real-valued $tt$.

$tri(t) = 1 + t if - 1 ≤ t ≤ 0 1 - t if 0 < t ≤ 1 0 otherwise tri(t) = 1 + t if - 1 ≤ t ≤ 0 1 - t if 0 < t ≤ 1 0 otherwise$