ELC 4351: Digital Signal Processing

Transcription

1 ELC 4351: Digital Signal Processing Liang Dong Electrical and Computer Engineering Baylor University liang October 18, 2016 Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

2 Frequency-domain Analysis of LTI Systems 1 Frequency-domain Characteristics of LTI Systems 2 Frequency Response of LTI Systems 3 Correlation Functions and Spectra at the Output of LTI Systems Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

3 Frequency-Domain Characteristics of Linear Time-Invariant Systems Response to Complex Exponential and Sinusoidal Signals: The Frequency Response Function H(ω) LTI system to an arbitrary input signal x(n): y(n) = h(k)x(n k) Input excitation is the complex exponential: x(n) = Ae jωn, < n <. The response is [ ] y(n) = h(k)ae jω(n k) = A h(k)e jωk e jωn Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

4 Frequency-Domain Characteristics of Linear Time-Invariant Systems [ y(n) = h(k)ae jω(n k) = A H(ω) = h(k)e jωk y(n) = AH(ω)e jωn h(k)e jωk ] e jωn x(n) = Ae jωn is an eigenfunction of the system. H(ω) is the corresponding eigenvalue of the system. Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

5 Frequency-Domain Characteristics of Linear Time-Invariant Systems Since H(ω) is the Fourier transform of {h(k)}, it follows that H(ω) is a periodic function with period 2π. Impulse Response and Frequency Response H(ω) = h(k) = 1 2π π π h(k)e jωk H(ω)e jωk dω Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

6 Frequency-Domain Characteristics of Linear Time-Invariant Systems If h(k) is a real-valued impulse response, then H R (ω) = H R ( ω) and H I (ω) = H I ( ω) H(ω) = H( ω) and H(ω) = H( ω) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

7 Response to Aperiodic Input Signals From the convolution theorem: LTI systems Y (ω) = H(ω)X (ω) Y (ω) = H(ω) X (ω) Y (ω) = H(ω) + X (ω) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

8 Response to Aperiodic Input Signals LTI systems Y (ω) = H(ω)X (ω) We observe that the output of a linear time-invariant system cannot contain frequency components that are not contained in the input signal. It takes either a linear time-variant system or a nonlinear system to create frequency components that are not necessarily contained in the input signal. Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

9 Response to Aperiodic Input Signals LTI systems Y (ω) = H(ω)X (ω) S yy (ω) = H(ω) 2 S xx (ω) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

10 Frequency Response of LTI Systems We focus on determining the frequency response of LTI systems that have rational system functions. Recall that this class of LTI systems is described in the time domain by constant-coefficient difference equations. Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

11 Frequency Response of a System with a Rational System Function If the system function H(z) converges on the unit circle, we can obtain the frequency response of the system by evaluating H(z) on the unit circle. H(ω) = H(z) z=e jω = If H(z) is a rational function, we have H(ω) = B(ω) A(ω) = n= h(n)e jωn M k=0 b ke jωk 1 + N k=1 a ke jωk Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

12 Frequency Response of a System with a Rational System Function H(ω) = M k=0 b ke jωk 1 + N k=1 a ke jωk = b 0 M k=1 (1 z ke jω ) N k=1 (1 p ke jω ) H (ω) = b 0 M k=1 (1 z k ejω ) N k=1 (1 p k ejω ) H (1/z ) = b 0 M k=1 (1 z k z) N k=1 (1 p k z) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

13 Frequency Response of a System with a Rational System Function When h(n) is real or, equivalently, the coefficients {a k } and {b k } are real, complex-valued poles and zeros occur in complex-conjugate pairs. In this case, H (1/z ) = H(z 1 ). Consequently, H (ω) = H( ω), and H(ω) 2 = H(ω)H (ω) = H(ω)H( ω) = H(z)H(z 1 ) z=e jω H(z)H(z 1 ) is the z-transform of r hh (n). It follows that H(ω) 2 is the Fourier transform of r hh (n). Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

14 Correlation Functions and Spectra at the Output of LTI Systems r yy (n) = r hh (n) r xx (n) r yx (n) = h(n) r xx (n) Using z-transform: S yy (z) = S hh (z)s xx (z) S yx (z) = H(z)S xx (z) Substitute z = e jω, we have S yy (ω) = H(ω) 2 S xx (ω) S yx (ω) = H(ω)S xx (ω) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

15 Input-Output Correlation Functions and Spectra The energy in the output signal: E y = 1 π 2π π = 1 π 2π π S yy (ω)dω = r yy (0) H(ω) 2 S xx (ω)dω Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

16 Correlation Functions and Power Spectra for Random Input Signals The expected value of the output signal: m y = E[y(n)] = E = = m x [ h(k)e[x(n k)] = m x = m x H(0) h(k) h(k)e jωk ω=0 h(k)x(n k) ] Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

17 Correlation Functions and Power Spectra for Random Input Signals The autocorrelation sequence of the output random process: γ yy (m) = E[y (n)y(n + m)] [ ] = E h(k)x (n k) h(l)x(n + m l) = = l= l= l= h(k)h(l)e[x (n k)x(n + m l)] h(k)h(l)γ xx (k l + m) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

18 Correlation Functions and Power Spectra for Random Input Signals The power density spectrum of the output random process: Γ yy (ω) = γ yy (m)e jωm = = m= m= [ l= = Γ xx (ω) l= [ = H(ω) 2 Γ xx (ω) h(k)h(l) h(k)h(l)γ xx (k l + m) [ m= l= ] e jωm γ xx (k l + m)e jωm ] ] [ ] h(k)e jωk h(l)e jωl Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

19 Correlation Functions and Power Spectra for Random Input Signals m y = m x H(0) Γ yy (ω) = H(ω) 2 Γ xx (ω) Γ yx (ω) = H(ω)Γ xx (ω) Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20

20 Correlation Functions and Power Spectra for Random Input Signals When the input random process is white, i.e., m x = 0 and γ xx (m) = σ 2 xδ(m) m y = 0 Γ xx (ω) = σx 2 Γ yy (ω) = H(ω) 2 σx 2 Γ yx (ω) = H(ω)σx 2 Liang Dong (Baylor University) Frequency-domain Analysis of LTI Systems October 18, / 20