Connexions

Site Feedback

Common Discrete Time Fourier Transforms

By Stephen Kruzick

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
Table 1: DTFTs
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