Lecture Outline ESE 531: Digital Signal Processing ! DT processing of CT signals " Impulse Invariance ! CT processing of DT signals (why??) Lec 9: February 9th, 2017 ! Downsampling CT Processing, Downsampling/Upsampling ! Upsampling Penn ESE 531 Spring 2017 - Khanna Penn ESE 531 Spring 2017 - Khanna 2 Discrete-Time Processing of Continuous Time Impulse Invariance ! Want to implement continuous-time system in x[n] y[n] discrete-time T T ! If xc(t) is bandlimited by π/T " I.e Xc(jΩ) = 0 for |Ω|> π/T ! then, ⎧ jω Yr ( jΩ) ⎪ H(e ) Ω < π / T = H ( jΩ) = ⎨ ω=ΩT X ( jΩ) eff c ⎩⎪ 0 else Penn ESE 531 Spring 2017 - Khanna 3 Penn ESE 531 Spring 2017 - Khanna 4 Impulse Invariance Impulse Invariance ! With Hc(jΩ) bandlimited, choose ! With Hc(jΩ) bandlimited, choose j ω j ω H(e ω ) = H ( j ), ω < π H(e ω ) = H ( j ), ω < π c T c T ! With the further requirement that T be chosen such ! With the further requirement that T be chosen such that that Hc ( jΩ) = 0, Ω ≥ π / T Hc ( jΩ) = 0, Ω ≥ π / T h[n] = Thc (nT ) Penn ESE 531 Spring 2017 - Khanna 5 Penn ESE 531 Spring 2017 - Khanna 6 1 Impulse Invariance Frequency Domain Analysis ! Let, h[n] = hc (nT ) ! If sampling at Nyquist Rate then ∞ j 1 ⎡ ⎛ω 2πk ⎞⎤ H(e ω ) H j = ∑ c ⎢ ⎜ − ⎟⎥ T k=−∞ ⎣ ⎝T T ⎠⎦ jΩT X s (e ) ω = ΩT Ω s ⋅T = π 2 Penn ESE 531 Spring 2017 - Khanna 7 Penn ESE 531 Spring 2017 - Khanna 8 Impulse Invariance Impulse Invariance ! Let, ! Let, Hc ( jΩ) = 0, Ω ≥ π / T Hc ( jΩ) = 0, Ω ≥ π / T h[n] = hc (nT ) h[n] =Th c (nT ) ! If sampling at Nyquist Rate then ! If sampling at Nyquist Rate then ∞ ∞ j 1 ⎡ ⎛ω 2πk ⎞⎤ j T1 ⎡ ⎛ω 2πk ⎞⎤ H(e ω ) H j H(e ω ) H j = ∑ c ⎢ ⎜ − ⎟⎥ = ∑ c ⎢ ⎜ − ⎟⎥ T k=−∞ ⎣ ⎝T T ⎠⎦ T k=−∞ ⎣ ⎝T T ⎠⎦ jω 1 ⎛ ω ⎞ jω T1 ⎛ ω ⎞ H(e ) = Hc ⎜ j ⎟, |ω|<π H(e ) = Hc ⎜ j ⎟, |ω|<π T ⎝ T ⎠ T ⎝ T ⎠ Penn ESE 531 Spring 2017 - Khanna 9 Penn ESE 531 Spring 2017 - Khanna 10 Impulse Invariance Example: DT Lowpass Filter ! Want to implement continuous-time system in ! We wish to implement a lowpass filter with cutoff discrete-time frequency Ωc on continuous time signal in discrete time with the following system h[n] = Thc (nT ) Penn ESE 531 Spring 2017 - Khanna 11 Penn ESE 531 Spring 2017 - Khanna 12 2 Impulse Invariance Continuous-Time Processing of Discrete-Time ! Want to implement continuous-time system in ! Useful to interpret DT systems with no simple discrete-time interpretation in discrete time h[n] = Thc (nT ) Penn ESE 531 Spring 2017 - Khanna 13 Penn ESE 531 Spring 2017 - Khanna 14 Continuous-Time Processing of Discrete-Time Continuous-Time Processing of Discrete-Time ⎧ j T ⎧ j T ⎪ TX (e Ω ) Ω < π / T ⎪ TX (e Ω ) Ω < π / T X c ( jΩ) = ⎨ X c ( jΩ) = ⎨ ⎩⎪ 0 else ⎩⎪ 0 else Also bandlimited Yc ( jΩ) = Hc ( jΩ)X c ( jΩ) Penn ESE 531 Spring 2017 - Khanna 15 Penn ESE 531 Spring 2017 - Khanna 16 Continuous-Time Processing of Discrete-Time Continuous-Time Processing of Discrete-Time jω 1 Y ( j ) H ( j )X ( j ) Y ( j ) H ( j )X ( j ) Y (e ) = Yc ( jΩ) c Ω = c Ω c Ω c Ω = c Ω c Ω T Ω=ω/T 1 ∞ 1 Y (e jω ) = Y ⎡ j Ω− kΩ ⎤ = Y ( jΩ) T ∑ c ⎣ ( s )⎦ T c k=−∞ Ω=ω/T Ω=ω/T Penn ESE 531 Spring 2017 - Khanna 17 Penn ESE 531 Spring 2017 - Khanna 18 3 Continuous-Time Processing of Discrete-Time Example jω 1 Y ( j ) H ( j )X ( j ) Y (e ) = Yc ( jΩ) c Ω = c Ω c Ω T Ω=ω/T 1 Y (e jω ) = H ( jΩ) X ( jΩ) T c Ω=ω/T c Ω=ω/T 1 = H ( jΩ) (TX (e jω )) ω < π T c Ω=ω/T H(e jω ) Penn ESE 531 Spring 2017 - Khanna 19 Penn ESE 531 Spring 2017 - Khanna 20 Example: Non-integer Delay Example: Non-integer Delay ! What is the time domain operation when Δ is non- ! What is the time domain operation when Δ is non- integer? I.e Δ=1/2 integer? I.e Δ=1/2 δ[n]↔1 − jωnd δ[n − nd ]↔ e Penn ESE 531 Spring 2017 - Khanna 21 Penn ESE 531 Spring 2017 - Khanna 22 Example: Non-integer Delay Example: Non-integer Delay ! The block diagram is for interpretation/analysis only yc (t) = xc (t −TΔ) Penn ESE 531 Spring 2017 - Khanna 23 Penn ESE 531 Spring 2017 - Khanna 24 4 Example: Non-integer Delay Example: Non-integer Delay ! The block diagram is for interpretation/analysis ! Delay system has an impulse response of a sinc with only a continuous time delay y[n] = y (nT ) = x (nt −TΔ) yc (t) = xc (t −TΔ) c c ⎛t − kT −TΔ ⎞ = x[k]sinc⎜ ⎟ ∑ T k ⎝ ⎠ t=nT = ∑x[k]sinc(n − k − Δ) k Penn ESE 531 Spring 2017 - Khanna 25 Penn ESE 531 Spring 2017 - Khanna 26 Example: Non-integer Delay Downsampling ! Delay system has an impulse response of a sinc with ! Similar to C/D conversion a continuous time delay " Need to worry about aliasing " Use anti-aliasing filter to mitigate effects ! If your discrete time signal is finely sample almost like a CT signal " Downsampling is just like sampling (C/D conversion) Penn ESE 531 Spring 2017 - Khanna 27 Penn ESE 531 Spring 2017 - Khanna 28 Downsampling Downsampling ! Definition: Reducing the sampling rate by an integer number Penn ESE 531 Spring 2017 - Khanna 29 Penn ESE 531 Spring 2017 - Khanna 30 5 Downsampling Downsampling jω jω ! Want to relate Xd(e ) to X(e ) not Xc(jΩ) ! k=rM+i " i = 0, 1, …, M-1 " r = -∞, ..., ∞ ! Separate sum into two sums—fine sum and coarse sum (i.e like counting minutes within hours) Penn ESE 531 Spring 2017 - Khanna 31 Penn ESE 531 Spring 2017 - Khanna 32 Downsampling Downsampling Penn ESE 531 Spring 2017 - Khanna 33 Penn ESE 531 Spring 2017 - Khanna 34 Downsampling Example Penn ESE 531 Spring 2017 - Khanna 35 Penn ESE 531 Spring 2017 - Khanna 36 6 Example Example 4π 2π 4π Penn ESE 531 Spring 2017 - Khanna 37 Penn ESE 531 Spring 2017 - Khanna 38 Example Example 6π 2π 6π Penn ESE 531 Spring 2017 - Khanna 39 Penn ESE 531 Spring 2017 - Khanna 40 Example Example 2π 4π 6π Penn ESE 531 Spring 2017 - Khanna 41 Penn ESE 531 Spring 2017 - Khanna 42 7 Example Upsampling ! Much like D/C converter ! Upsample by A LOT # almost continuous ! Intuition: " Recall our D/C model: x[n] # xs(t)#xc(t) " Approximate “xs(t)” by placing zeros between samples " Convolve with a sinc to obtain “xc(t)” Penn ESE 531 Spring 2017 - Khanna 43 Penn ESE 531 Spring 2017 - Khanna 44 Upsampling Upsampling ! Definition: Increasing the sampling rate by an integer number x[n] = xc (nT ) xi[n] = xc (nT ') xi[n] Penn ESE 531 Spring 2017 - Khanna 45 Penn ESE 531 Spring 2017 - Khanna 46 Upsampling Frequency Domain Interpretation Penn ESE 531 Spring 2017 - Khanna 47 Penn ESE 531 Spring 2017 - Khanna 48 8 Frequency Domain Interpretation Frequency Domain Interpretation Penn ESE 531 Spring 2017 - Khanna 49 Penn ESE 531 Spring 2017 - Khanna 50 Frequency Domain Interpretation Example Penn ESE 531 Spring 2017 - Khanna 51 Penn ESE 531 Spring 2017 - Khanna 52 Example Example Penn ESE 531 Spring 2017 - Khanna 53 Penn ESE 531 Spring 2017 - Khanna 54 9 Example Example Penn ESE 531 Spring 2017 - Khanna 55 Penn ESE 531 Spring 2017 - Khanna 56 Example Big Ideas ! DT processing of CT signals " Impulse Invariance to design DT systems for CT signals ! CT processing of DT signals " Allows for interpretation of DT systems ! Downsampling " Like a C/D converter ! Upsampling " Like a D/C converter Penn ESE 531 Spring 2017 - Khanna 57 Penn ESE 531 Spring 2017 - Khanna 58 Admin ! HW 3 due Friday ! HW 4 posted after class Penn ESE 531 Spring 2017 - Khanna 59 10 .