Spectral Analysis of Sinusoidal Signals In practice,the infinite-length sequence g[n]is first windowed by multiplying it with a length-N window wIn to convert it into a length-N sequence y n] DTFT T(ei)of y[n]then is assumed to provide a reasonable estimate of G(ei) T(ei)is evaluated at a set of R(R>N) discrete angular frequencies equally spaced in the range 0<o<2n by computing the R-point FFT I(k)of y[n]
Spectral Analysis of Sinusoidal Signals • In practice, the infinite-length sequence g[n] is first windowed by multiplying it with a length-N window w[n] to convert it into a length-N sequence γ[n] • DTFT Γ(ejω) of γ[n] then is assumed to provide a reasonable estimate of G(ejω) • Γ(ejω) is evaluated at a set of R ( R≥N) discrete angular frequencies equally spaced in the range 0≤ω≤2π by computing the R-point FFT Γ(k) of γ[n]
Spectral Analysis of Sinusoidal Signals We analyze the effect of windowing and the evaluation of the frequency samples of the DTFT via the DFT Now IIk]-T(el@)o-2kR 0≤k≤R-1 The normalized discrete-time angular frequency ok corresponding to the DFT bin number k(DFT frequency)is given by 2πk 0k= R
Spectral Analysis of Sinusoidal Signals • We analyze the effect of windowing and the evaluation of the frequency samples of the DTFT via the DFT [ ] ( ) , 0 1 2 / Γ = Γ ≤ ≤ − = k e k R k R j ω π ω R k k π ω 2 = • The normalized discrete-time angular frequency ωk corresponding to the DFT bin number k (DFT frequency) is given by Now
Spectral Analysis of Sinusoidal Signals The continuous-time angular frequency corresponding to the DFT bin number k (DFT frequency)is given by 2k= 2πk RT To interpret the results of DFT-based spectral analysis correctly we first consider the frequency-domain analysis of a sinusoidal signal
Spectral Analysis of Sinusoidal Signals • The continuous-time angular frequency corresponding to the DFT bin number k (DFT frequency) is given by RT k k 2π Ω = • To interpret the results of DFT-based spectral analysis correctly we first consider the frequency-domain analysis of a sinusoidal signal
Spectral Analysis of Sinusoidal Signals Consider g[n]=cos(0on+φ)，-o<n<o It can be expressed as gl川=ea,n+)+eon+） Its DTFT is given by G(ejo)=n∑ej06(o-0+2π) =-00 00 +π ej06(o-oo+2π =-00
Spectral Analysis of Sinusoidal Signals Its DTFT is given by g[n] = cos(ωon +φ), − ∞ < n < ∞ ( ) ( ) ( ) 2 1 [ ] ω +φ − ω +φ = + j n j n g n e o e o = ∑ − + ∞ =−∞ ( ) π δ (ω ω 2π) ω φ o j j G e e + ∑ − + ∞ =−∞ − π δ (ω ω 2π) φ o j e Consider It can be expressed as
Spectral Analysis of Sinusoidal Signals G(ei)is a periodic function of o with a period 2n containing two impulses in each period In the range-r≤ω≤r,there is an impulse at ω=ωo of complex amplitude元eiφand an impulse atω=-ωo of complex amplitudeπe-iφ To analyze g[n using DFT,we employ a finite- length version of the sequence given by y[n]=cos(oon+p),0≤n≤W-1
Spectral Analysis of Sinusoidal Signals • G(ejω) is a periodic function of ω with a period 2π containing two impulses in each period • In the range -π≤ω≤π , there is an impulse at ω=ω0 of complex amplitude πejφ and an impulse at ω=-ω0 of complex amplitude πe-jφ • To analyze g[n] using DFT, we employ a finitelength version of the sequence given by γ [n] = cos(ωon +φ), 0 ≤ n ≤ N −1