ECE 463: Digital Communications Lab. Lecture 3: Pulse Shaping and Matched Filters Haitham Hassanieh Previous Lecture: ü Up Conversion & Down Conversion ü PAM vs QAM Spectral Efficiency ü Software Defined Radios This Lecture: q Pulse Shaping Filters q Matched Filters q Symbol Timing Recovery q Eye Diagrams Digital Communication System 1011010110011001 1011010110011001 TX RX Bits Mixer Bits-to-Symbols PA Mapper Modulation Pulse DAC LPF BPF (Encoding) Shaping PLL Bits Mixer Symbols-to-Bits Mapper LNA BPF LPF ADC Channel Demodulation Equalization (Decoding) & Synchronization PLL Digital Communication System Mixer Transmitter ℜ'{ } LPF BPF Bits DAC PLL Bits-to-Symbols PA Mapper + 90% Modulation Pulse (Encoding) Shaping ℑ+{ } DAC LPF Mixer BPF Mixer Receiver BPF LPF ADC , Bits PLL Symbols-to-Bits + Mapper % LNA 90 Channel Demodulation Equalization (Decoding) & -. Synchronization BPF Mixer LPF ADC Digital Communication System 1011010110011001 1011010110011001 TX RX Bits Mixer Bits-to-Symbols PA Mapper Modulation Pulse DAC LPF BPF (Encoding) Shaping PLL Bits Mixer Symbols-to-Bits Mapper LNA BPF LPF ADC Channel Demodulation Equalization (Decoding) & Synchronization PLL Digital Communication System 1011010110011001 1011010110011001 TX RX Bits Mixer Bits-to-Symbols PA Mapper Modulation Pulse DAC LPF BPF (Encoding) Shaping PLL Bits Mixer Symbols-to-Bits Mapper LNA BPF LPF ADC Channel Demodulation Equalization (Decoding) & Synchronization PLL Digital Communication System 1011010110011001 1011010110011001 TX RX Bits Mixer Bits-to-Symbols PA Mapper Modulation Pulse DAC LPF BPF (Encoding) Shaping PLL Bits Mixer Symbol Timing Symbols-to-Bits Recovery Mapper LNA BPF LPF ADC Matched Channel Demodulation Filter Equalization (Decoding) & Synchronization PLL PAM: Pulse Amplitude Modulation 11 10 00 01 −3 −1 +1 +3 & ' :1011000110011001 Bits-to-Symbols Mapper ) ' :-1,-3,+1,+3,-1,+1,-1 Modulation (Encoding) 12 * + = - ) ' . + − '/0 3452 • /0: Symbol Time/ Baud Time 8 • 70 = : Symbol Rate or Baud Rate 9: • . + : pulse shape Pulse Shaping 11 10 00 01 −3 −1 +1 +3 & ' :1011000110011001 +3 * + Bits-to-Symbols ) ' :-1,-3,+1,+3,-1,+1,-1 Mapper +1 + Modulation (Encoding) 12 −1 * + = - ) ' . + − '/0 −3 3452 • Simplest pulse shape: Rectangle . + = Π 8 . + 9: 1 + /0 Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* !" 1 # , !" 1 1 − !" !" Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* !" 1 1/, # , !" 1 1 − !" !" Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* 1 1/, # , !" 1 1 − !" !" + , 4 Very wide bandwidth! ≈ −1378 Low spectral efficiency , 1 1 − !" !" Pulse Shaping: Rectangular Pulse Satisfies the Nyquist Criterion For Inter-Symbol-Interference +3 +1 ! −1 −3 Sampled Values: & ' = −1, −3, +1, +3, −1, +1, −1 … & ' = ,['] No Inter-Symbol-Interference (ISI) ISI: & ' = , ' + / 01 , ' − 1 + / 201 , ' − 2 + … Rectangular Pulse No ISI Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* 1 1/, # , !" 1 1 − !" !" + , 4 Very wide bandwidth! ≈ −1378 Low spectral efficiency No Inter-Symbol-Interference , 1 1 − !" !" Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* 1 1/, # , !" 1 1 − !" !" + , 4 Very wide bandwidth! ≈ −1378 Low spectral efficiency No Inter-Symbol-Interference , 1 1 − !" !" Pulse Shaping: Rectangular Pulse ( % # = Π + , = !"sinc 1!", )* 1 1/, # , !" 1 1 − !" !" 46( % # ∝ sinc 4 )* + , # , 1 1 − !" !" Pulse Shaping: Rectangular Pulse '/0 ) ( ∝ sinc ' 12 & ! ( ! 1 1 − $% $% Pulse Shaping: Rectangular Pulse '/0 ) ( ∝ sinc ' 12 & ! ( ! 1 1 − $% $% Infinite Responseà Impossible to realize in practice High ISI if sampling is not perfect No ISI if sampling is aligned Bandlimited in Frequency Pulse Shaping: Raised Cosine Pulse Shaping: Raised Cosine 1 − $ &' ! ≤ 2&' ( ! = & 0& 1 − $ 1 − $ 1 + $ ' 1 + cos ' ! − ≤ ! ≤ 2 $ 2&' 2&' 2&' 0 otherwise ( ! &' ! 1 − $ 1 − $ − 2&' 2&' Pulse Shaping: Raised Cosine 1 − $ &' ! ≤ 2&' ( ! = & 0& 1 − $ 1 − $ 1 + $ ' 1 + cos ' ! − ≤ ! ≤ 2 $ 2&' 2&' 2&' 0 otherwise ( ! &' ! 1 + $ 1 − $ 1 − $ 1 + $ − − 2&' 2&' 2&' 2&' Pulse Shaping: Raised Cosine 1 − $ &' ! ≤ 2&' ( ! = & 0& 1 − $ 1 − $ 1 + $ ' 1 + cos ' ! − ≤ ! ≤ 2 $ 2&' 2&' 2&' 0 otherwise ( ! &' &'/2 ! 1 + $ −1 1 − $ 1 − $ 1 1 + $ − − 2&' 2&' 2&' 2&' 2&' 2&' Pulse Shaping: Raised Cosine 1 − $ &' ! ≤ 2&' ( ! = & 0& 1 − $ 1 − $ 1 + $ ' 1 + cos ' ! − ≤ ! ≤ 2 $ 2&' 2&' 2&' 0 otherwise ( ! &' &'/2 ! 1 + $ −1 1 − $ 1 − $ 1 1 + $ − − 2&' 2&' 2&' 2&' 2&' 2&' Pulse Shaping: Raised Cosine 1 − * $% " ≤ 2$% ! " = $ 0$ 1 − * 1 − * 1 + * % 1 + cos % " − ≤ " ≤ 2 * 2$% 2$% 2$% 0 otherwise ! " ! " 9 $% $%/2 " " 1 + * −1 1 − * 1 − * 1 1 + * 1 + * −1 1 − * 1 − * 1 1 + * − − − − 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% Pulse Shaping: Raised Cosine 1 − * $% " ≤ 2$% ! " = $ 0$ 1 − * 1 − * 1 + * % 1 + cos % " − ≤ " ≤ 2 * 2$% 2$% 2$% 0 otherwise ! " ! " 9 $% −3;< cut-off $%/2 " " 1 + * −1 1 − * 1 − * 1 1 + * 1 + * −1 1 − * 1 − * 1 1 + * − − − − 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% Pulse Shaping: Raised Cosine 1 − * $% " ≤ 2$% ! " = $ 0$ 1 − * 1 − * 1 + * % 1 + cos % " − ≤ " ≤ 2 * 2$% 2$% 2$% 0 otherwise ! " ! " 9 $% −3;< cut-off $%/2 " " 1 + * −1 1 − * 1 − * 1 1 + * 1 + * −1 1 − * 1 − * 1 1 + * − − − − 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 1 $% Pulse Shaping: Raised Cosine 1 − * $% " ≤ 2$% ! " = $ 0$ 1 − * 1 − * 1 + * % 1 + cos % " − ≤ " ≤ 2 * 2$% 2$% 2$% 0 otherwise ! " ! " 9 $% −3;< cut-off $%/2 " " 1 + * −1 1 − * 1 − * 1 1 + * 1 + * −1 1 − * 1 − * 1 1 + * − − − − 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 2$% 1 * 2$ $% % Pulse Shaping: Raised Cosine ' ( !" !"/2 ( −1 1 2!" 2!" Pulse Shaping: Raised Cosine ' ( ) = 0.5 !" !"/2 ( −1 1 2!" 2!" 1 !" ( ≤ 4!" ' ( = !" 5 1 3 1 + cos 25!" ( − ≤ ( ≤ 2 2 4!" 4!" 0 otherwise Pulse Shaping: Raised Cosine ' ( ) = 0.5 !" ) = 0.25 !"/2 ( −1 1 2!" 2!" 3 !" ( ≤ 8!" ' ( = !" 37 3 5 1 + cos 47!" ( − ≤ ( ≤ 2 2 8!" 8!" 0 otherwise Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ( −1 1 2!" 2!" !" 1 ' ( = 1 + cos 3!" ( ( ≤ 2 !" 0 otherwise Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" 1 !" ( ≤ 2!" ' ( = 0 otherwise Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" 1 − ) !" ( ≤ 2!" ' ( = ! 4! 1 − ) 1 − ) 1 + ) " 1 + cos " ( − ≤ ( ≤ 2 ) 2!" 2!" 2!" 0 otherwise ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" 1 − ) !" ( ≤ 2!" ' ( = ! 4! 1 − ) 1 − ) 1 + ) cos 4)</! " 1 + cos " ( − ≤ ( ≤ " ; < = sinc </!" ? ? ? 2 ) 2!" 2!" 2!" 1 − 4 ) < /!" 0 otherwise ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" . / cos 5)//!" . / = sinc //!" 8 8 8 1 − 4 ) / /!" / ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" . / cos 5)//!" . / = sinc //!" 8 8 8 1 − 4 ) / /!" / ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" . / cos 5)//!" . / = sinc //!" 8 8 8 1 − 4 ) / /!" / ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" . / cos 5)//!" . / = sinc //!" 8 8 8 1 − 4 ) / /!" / ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" 2 3 cos 9)3/!" 2 3 = sinc 3/!" < < < 1 − 4 ) 3 /!" 3 ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage ↓ ) ∶ ↓ Bandwidth leakage Pulse Shaping: Raised Cosine ) = 1 ' ( ) = 0.5 !" ) = 0.25 !"/2 ) = 0 ( −1 1 2!" 2!" 3 4 cos :)4/!" 3 4 = sinc 4/!" < < < 1 − 4 ) 4 /!" 4 ): Rolloff Factor ↑ ) ∶ ↑ Bandwidth leakage, ↓ Time Support, ↓ Sidelobes à ↓ ISI if sampling not aligned ↓ ) ∶ ↓ Bandwidth leakage, ↑ Time Support, ↑ Sidelobes à ↑ ISI if sampling not aligned Digital Communication System 1011010110011001 1011010110011001 TX RX Bits Mixer Bits-to-Symbols PA Mapper Modulation Pulse DAC LPF BPF (Encoding) Shaping PLL Bits Mixer Symbol Timing Symbols-to-Bits Recovery Mapper LNA BPF LPF ADC Matched Channel Demodulation Filter Equalization (Decoding) & Synchronization PLL Digital Communication System 1011010110011001 1011010110011001 TX RX Pulse Shaping Matched Filter Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Pulse Matched Shaping Filter Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter *+ # $ 0 & = " # $ % & − $() ,-.+ Let us focus on a single symbol i.e., single $ Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter " & + # = " # ∗ % # " # = " & ' # − &)* Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter " ' , # = " # ∗ %& # " # = " ' ( # − '*+ Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter * . $ " = * " ∗ ,- " ! " = $ " + &(") Consider Simple AWGN Channel. &(") is Gaussian noise. Pulse Shaping and Matched Filtering 1011010110011001 1011010110011001 TX RX Channel Pulse Matched Shaping Filter !. " = ! " ∗ + (") ) - $ " = ) " ∗ +, " ! " = $ " + &(") / !. " = ) - 0 " − -23 ∗ +, " ∗ +/ " + &