Lecture Outline

ESE 531: Digital ! DT processing of CT signals " Impulse Invariance

! CT processing of DT signals (why??) Lec 9: February 9th, 2017 ! Downsampling CT Processing, Downsampling/ ! Upsampling

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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

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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 )

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1 Impulse Invariance Frequency Domain Analysis

! Let,

h[n] = hc (nT ) ! If sampling at 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 ⎠

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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)

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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

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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

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Example: Non-integer Delay Example: Non-integer Delay

! The block diagram is for interpretation/analysis only

yc (t) = xc (t −TΔ)

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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

" 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)

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Downsampling Downsampling

! Definition: Reducing the sampling rate by an integer number

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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)

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Downsampling Downsampling

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Downsampling Example

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6 Example Example

4π 2π 4π

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Example Example

6π 2π 6π

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Example Example

2π 4π 6π

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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)”

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Upsampling Upsampling

! Definition: Increasing the sampling rate by an integer number

x[n] = xc (nT )

xi[n] = xc (nT ')

xi[n]

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Upsampling Frequency Domain Interpretation

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8 Frequency Domain Interpretation Frequency Domain Interpretation

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Frequency Domain Interpretation Example

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Example Example

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9 Example Example

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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

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Admin

! HW 3 due Friday ! HW 4 posted after class

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