电子科技大学：《数字信号处理 Digital Signal Processing》课程教学资源（课件讲稿）Chapter 11 Applications of Digital Signal Processing

Chapter 11 Applications of Digital Signal Processing

Chapter 11 Applications of Digital Signal Processing

Spectral Analysis of Signals Spectral analysis is concerned with the determination of frequency contents of a continuous-time signal ga(t)using DSP methods It involves the determination of either the energy spectrum or the power spectrum of the signal If ga(t)is sufficiently bandlimited,spectral characteristics of its discrete-time equivalent g[n]should provide a good estimate of spectral characteristics of ga(t)

Spectral Analysis of Signals • Spectral analysis is concerned with the determination of frequency contents of a continuous-time signal ga(t) using DSP methods • It involves the determination of either the energy spectrum or the power spectrum of the signal • If ga(t) is sufficiently bandlimited, spectral characteristics of its discrete-time equivalent g[n] should provide a good estimate of spectral characteristics of ga(t)

Spectral Analysis of Signals In most cases,ga(t)is defined for-oo<t<oo Thus,gn is of infinite extent,and defined for -co<n<oo Hence,ga(t)is first passed through an analog anti-aliasing filter whose output is then sampled to generate gn Assumptions:(1)Effect of aliasing can be ignored,(2)A/D conversion noise can be neglected

Spectral Analysis of Signals • In most cases, ga(t) is defined for-∞<t<∞ • Thus, g[n] is of infinite extent, and defined for -∞<n<∞ • Hence, ga(t) is first passed through an analog anti-aliasing filter whose output is then sampled to generate g[n] • Assumptions: (1) Effect of aliasing can be ignored, (2) A/D conversion noise can be neglected

Spectral Analysis of Signals Three types of spectral analysis- 1)Spectral analysis of stationary sinusoidal signals 2)Spectral analysis of of nonstationary signals with time-varying parameters 3)Spectral analysis of random signals

Spectral Analysis of Signals • Three types of spectral analysis - • 1) Spectral analysis of stationary sinusoidal signals • 2) Spectral analysis of of nonstationary signals with time-varying parameters • 3) Spectral analysis of random signals

Spectral Analysis of Sinusoidal Signals Assumption -Parameters characterizing sinusoidal signals,such as amplitude, frequencies,and phase,do not change with time For such a signal g[n],the Fourier analysis can be carried out by computing the DTFT G(ejo)=∑g[nle-jon n=-o∞

Spectral Analysis of Sinusoidal Signals • Assumption - Parameters characterizing sinusoidal signals, such as amplitude, frequencies, and phase, do not change with time • For such a signal g[n], the Fourier analysis can be carried out by computing the DTFT = ∑ ∞ =−∞ − n j j n G e g n e ω ω ( ) [ ]

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 finite￾length version of the sequence given by γ [n] = cos(ωon +φ), 0 ≤ n ≤ N −1