Signal Functions (0B)

Signal Functions (0B) Signal Functions

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Sinc Function Properties sinc(t) = sin(πt) π t sinc(t) = sin(πt) π t * sinc ( t) = sin ( πt) πt = sinc(t) lim t 0 sinc(t) = lim t 0 π cos(πt) π = an even function Maximum: sinc (0) = when t = 0 t = 0 sinc (0) = lim t 0 sinc(t) = lim t 0 π cos(πt) π = t = n sinc(nπ) = 0 n: integer (n 0) sinc(t) = sin(nπ) nπ = 0 3

Sinc Function Properties 2 π t.5 sin(πt) 0.5 0-0.5 - -.5-2 -6-4 -2 0 2 4 6 4

Sinc Function Properties 2 π t.5 sin(πt) 0.5 sin(π t) πt 0-0.5 - -.5-2 -6-4 -2 0 2 4 6 5

Sinc Function Properties sin(π t) πt 0.8 0.6 0.4 0.2 0-0.2-0.4-6 -4-2 0 2 4 6 6

Dirichlet Function () Dirichlet Function drcl(t, L) = sin(π Lt ) Lsin(π t) t = m L sin(π Lt ) = 0 m: integer multiples of L t n sin(πt ) 0 drcl(t, L) = sin(π Lt ) Lsin(π t) = 0 odd L lim t n cos(π Lt ) cos(πt) = + t = n lim t n drcl (t, L) = sin(πt ) = 0 (sin(π L t))' ( Lsin (π t))' n: integer = lim t n L π cos(π L t) L π cos(π t) = ± drcl (n, L) = + n = 2 n = n = 0 n = + n = +2 n = +3 even L lim t n cos(π Lt ) cos(πt) = ( ) n drcl(n, L) = ( ) n cos( L 2 π) cos( 2 π) + + cos( L π), cos( π), cos(0) cos( L π) cos(l 2π) cos( L 3π),,, cos(0) cos(π) cos(2π) cos(3π), + + + + + + + odd L even L 7

Dirichlet Function (2) Dirichlet Function drcl(t, L) = sin(π Lt ) Lsin(π t) odd L π t ω/2 D L (e j ω ) = sin( ω L/2) Lsin( ω/2) ω x even L diric( x, N ) = sin ( N x/2) N sin( x/2) 8

Dirichlet Function Properties Dirichlet Function drcl(t, L) = sin(π Lt ) Lsin(π t) diric( x, N ) = sin ( N x/2) N sin( x/2) D L (e j ω ) = sin( ω L/2) Lsin( ω/2) D L (e j( ω + 2π) ) = = sin(( ω + 2π) L/2) Lsin (( ω + 2π)/2) sin( ω L/2 + L π) Lsin( ω/2 + π) 0 ω +π 0 ω/2 + π 2 0 ω L/2 +L π 2 0 sin( ω/2) + sin( ω L/2) + +D L (e j ω ) for an odd L D L (e j ω ) for an even L (period: 2π) a quarter period Envelope: sin( ω/2) L quarter periods Zeros: ω = 2π L k sin( ω L/2) = 0 D L (e j ω ) = sin ( ω L/2) Lsin ( ω/2) = D L(e j ω ) lim D L (e j ω ) = lim ω 0 ω 0 L/2cos( ω L/2) L/2cos( ω/2) = an even function Maximum: D L (e j ω ) = when ω = 0 9

A Dirichlet Function L=9 () 3 D 9 (e j ω ) = sin( ω9/2) 9sin ( ω/2) 2 sin( ω9/2) sin( ω/2) 0 - mutliplication -2-3 0 2 4 6 8 0 2 sin( ω/2) becomes an envelope ω = 0 ω = π ω = 2π ω = 3π ω = 4 π 0

A Dirichlet Function L=9 (2) 5 lim D 9 (e j ω ) = ω 0 9 0 5 sin( ω9/2) sin( ω/2) Envelope: sin( ω/2) D 9 (e j ω ) = ω = π ω 2 = π 2 ω9 2 = 9 π 2 sin( ω9/2) 9sin ( ω/2) a quarter cycle 9 quarter cycles sin( ω9/2) = 0 Zeros 0 2 3 4 5 6 7 8 9 sin( ω9/2) -5 0 0.5.5 2 2.5 3 ω = 0 9 quarter cycles ω = π ω9/2 = k π lim D L (e j ω ) ω 0 (sin( ω9/ 2))' lim ω 0 (9sin ( ω/2))' ω = k 2 π 9 Maximum = approaches to the unit Zeros: ω = 2π 9 k

A Dirichlet Function L=9 (3) an even function D L (e j ω ) = sin( ω L /2) L sin( ω/2) = D L(e j ω ) symmetric along the y axis D 9 (e j ω ) = sin( ω9/2) 9sin ( ω/2) ω = π π ω = 0 ω = +π π D L (e j( ω + 2 π) ) = sin( ω L/2 + L π) L sin( ω/2 + π) = +D L (e j ω ) for an odd L D L (e j ω ) for an even L (period: 2π) 2

A Dirichlet Function L=9 (4) D L (e j ω ) = sin( ω L /2) L sin( ω/2) = D L(e j ω ) D 9 (e j ω ) = sin( ω9/2) 9sin ( ω/2) symmetric symmetric symmetric ω = π ω = 0 ω = +π ω = 2π ω = 3 π π π π π D L (e j( ω + 2 π) ) = sin( ω L/2 + L π) L sin( ω/2 + π) = +D L (e j ω ) for an odd L D L (e j ω ) for an even L (period: 2π) 3

Dirichlet Functions (L: Odd) D 9 (e j ω ) 2 π D 9 (e j ω ) = sin( ω9/2) 9sin ( ω/2) 8 zero crossings D (e j ω ) = sin ( ω/2) sin( ω/2) 0 zero crossings 8 zero crossings D 3 (e j ω ) = sin ( ω3/2) 3sin( ω/2) 2 zero crossings D (e j ω ) 2 π D 3 (e j ω ) 2 π 0 zero crossings zero crossings 4

Dirichlet Functions (L: Even) D 0 (e j ω ) 2 π D 0 (e j ω ) = sin( ω0/2) 0sin ( ω/2) 9 zero crossings 9 zero crossings D 2 (e j ω ) = D 4 (e j ω ) = sin ( ω2/2) 2sin( ω/2) sin( ω4/2) 4 sin ( ω/2) zero crossings 3 zero crossings D 2 (e j ω ) 2 π D 4 (e j ω ) 2 π zero crossings 3 zero crossings 5

References [] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] G. Beale, http://teal.gmu.edu/~gbeale/ece_220/fourier_series_02.html [4] C. Langton, http://www.complextoreal.com/chapters/fft.pdf