Spectral Analysis of Sinusoidal Signals Since fR 11×32 =5.5 FT 64 the impulse at f=11 Hz of the DTFT appear between the DFT bin locations k=5 and k=6 Likewise,the impulse f=-11 Hz of the DTFT appear between the DFT bin locations k=26 and k=27
Spectral Analysis of Sinusoidal Signals • the impulse at f = 11 Hz of the DTFT appear between the DFT bin locations k = 5 and k = 6 • Likewise, the impulse f = -11 Hz of the DTFT appear between the DFT bin locations k = 26 and k = 27 5.5 64 11 32 = × = FT f R Since
Simusuida s DFT magnitude plot 15 10 5 0 PPPPPPPePPRiP1 10 20 30 k Note:Spectrum contains frequency components at all bins,with two strong components at k=5 and k=6,and two strong components at k=26 and k=27
Spectral Analysis of Sinusoidal Signals • Note: Spectrum contains frequency components at all bins, with two strong components at k = 5 and k = 6, and two strong components at k = 26 and k = 27 •DFT magnitude plot
Spectral Analysis of Sinusoidal Signals The phenomenon of the spread of energy from a single frequency to many DFT frequency locations is called leakage To understand the cause of leakage,recall that the N-point DFT G[k]of a length-N sequence g[n]is given by the samples of its DTFT T(ei): r]=rea-2k0≤k≤N-
Spectral Analysis of Sinusoidal Signals • The phenomenon of the spread of energy from a single frequency to many DFT frequency locations is called leakage • To understand the cause of leakage, recall that the N-point DFT G[k] of a length-N sequence g[n] is given by the samples of its DTFT Γ(ejω) : [ ] ( ) , 0 1 2 / Γ = Γ ≤ ≤ − = k e k N k N j k k ω π ω
Spectral Analysis of Sinusoidal Signals Plot of the DTFT of the length-32 sinusoidal sequence of frequency 11 Hz sampled at 64 Hz is shown below along with its 32-point DFT 15 T()l 10 5 a9a08ae@$99 10 20 30 k
Spectral Analysis of Sinusoidal Signals • Plot of the DTFT of the length-32 sinusoidal sequence of frequency 11 Hz sampled at 64 Hz is shown below along with its 32-point DFT
Spectral Analysis of Sinusoidal Signals The DFT samples are indeed obtained by the frequency samples of the DTFT Now the sequence y[n]=cos(oon+p),0≤n≤N-1 has been obtained by windowing the infinite- length sinusoidal sequence gn]with a rectangular window w[n: 0≤n≤W-1 otherwise
Spectral Analysis of Sinusoidal Signals has been obtained by windowing the infinitelength sinusoidal sequence g[n] with a rectangular window w[n]: γ [n] = cos(ωon +φ), 0 ≤ n ≤ N −1 ≤ ≤ − = 0, otherwise 1, 0 1 [ ] n N w n The DFT samples are indeed obtained by the frequency samples of the DTFT Now the sequence