Filtering in Continuous and Discrete Time

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1 Filtering in Continuous and Discrete Time Lowpass, highpass, bandpass iltering Filter response to cosine wave inputs Discrete-Time Fourier Transorm Filtering based on dierence equations Copyright 27 by M.H. Perrott ll rights reserved. M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide

2 Motivation or Filtering RF Transmission RF Reception Voice RF Transmitter Intererer RF Receiver Voice X() Y() Voiceband requencies RF transmission requencies Voiceband requencies RF reception requencies Filtering is used to remove undesired signals outside o the requency band o interest Enables selection o a speciic radio, TV, WLN, cell phone, cable TV channel Undesired channels are oten called intererers M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 2

3 Lowpass Filter z(t) w(t) Lowpass Filter H() r(t) y(t) = 2cos(2π o t) 2 W() H() -2 o - o o 2 o -2 o - o o 2 o Only low requency (i.e. baseband) portion remains -2 o R() 2 - o o 2 o M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 3

4 Highpass Filter z(t) w(t) Highpass Filter H() r(t) y(t) = 2cos(2π o t) 2 W() H() -2 o - o o 2 o -2 o - o o 2 o R() Only high requency (i.e. RF) portion remains -2 o - o o 2 o M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 4

5 Bandpass Filter z(t) w(t) Bandpass Filter H() r(t) y(t) = 2cos(2π o t) 2 W() H() -2 o - o o 2 o -2 o - o o 2 o R() Only high requency (i.e. RF) portion remains -2 o - o o 2 o M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 5

6 Why is Bandpass Filtering Useul? llows removal o interering signals Highpass iltering would be o limited use here Typically higher complexity implementation than with lowpass or highpass ilters Many RF systems such as cell phones use specialized components called SW ilters to achieve bandpass iltering 2 W() Intererer H() -2 o - o o 2 o -2 o - o o 2 o R() Only desired RF portion remains Intererer eliminated -2 o - o o 2 o M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 6

7 More Formal Treatment o Filters Ideal Lowpass Practical Lowpass K H() K H() 2 K - c c - c c H() H() - c c - c c n ideal ilter would have a brickwall magnitude response and zero phase response Practical ilters have a more gradual magnitude rollo and a non-zero phase response Design o the ilter usually ocuses on getting a reasonable magnitude rollo with a speciied cuto requency c (i.e., ilter bandwidth) M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 7

8 Response o Filter to Input Cosine Lowpass Filter: H() - o X() o K - o H() o H() H(- o ) e j - o H(- o ) Y() o H( o ) e j H( o ) - o o Fourier transorm analysis: Key properties o practical ilters Magnitude response is even: Phase response is odd: We ll explain why this is true in 6.3 M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 8

9 Compute Fourier Transorm Lowpass Filter: H() - o X() o K - o H() o H() H(- o ) e j - o H(- o ) Y() o H( o ) e j H( o ) - o o Deine: Fourier transorm o output: M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 9

10 Compute Time-Domain Response Lowpass Filter: H() - o X() o K - o H() o H() H(- o ) e j - o H(- o ) Y() o H( o ) e j H( o ) - o o Transorm back to time domain: M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide

11 Key Observations o Filter Response Lowpass Filter: H() - o X() o K - o H() o H() H(- o ) e j - o H(- o ) Y() o H( o ) e j H( o ) - o o Input cosine wave is scaled in amplitude and phase-shited in time Scale actor set by magnitude o H() at = o Phase shit set by phase o H() at = o We typically ocus only on the magnitude o the requency response, H(), o the ilter M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide

12 Designing and Using Filters Within Matlab Real World Matlab x(t) x[n] t Our lab exercises will have you design and use ilters in Matlab Matlab will interace to the USRP board in order to receive real world signals rom the antenna Matlab ramework is based on discrete-time sequences (which are indexed on integer values) Correspond to samples o corresponding real world signals (which are continuous-time in nature) n We need another Fourier analysis tool M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 2

13 The Discrete-Time Fourier Transorm llows us to deal with non-periodic, discrete-time signals Frequency domain signal is periodic in this case x[n] Where: n X(e j2πλ ) - λ Note: t unction in Matlab used to compute DTFT M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 3

14 Relating to Samples o `Real World Signals Matlab Sequence Samples o Real World Signal x[n] x(t) n T t X(e j2πλ ) X() - λ Samples o a continuous-time signal with sample period T leads to requency domain signal with period /T -/T We simply scale requency axis o t in Matlab We will say much more about sampling later /T M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 4

15 Filters Within Matlab x[n] Discrete-Time Filter Implemented as dierence equations x[n] n H(e j2πλ ) Current output, y[n], depends on weighted values o previous output samples and current and previous input samples, x[n] y[n] y[n] n Group a and b coeicients as vectors: Execute ilter using the ilter command: M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 5

16 Impact o Delay on DTFT Consider a signal that is a delayed version o another signal: Compute DTFT o y[n] M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 6

17 Compute Filter Response using DTFT Make use o the time shit property: Filter response is simply ratio o output over input: M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 7

18 FIR Filters Finite Impulse Response (FIR) ilters use only b coeicients in their implementation Example: x[n] H(e j2πλ ) y[n] Note: a k b k M k N k M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 8

19 x[n] H(e j2πλ ) Filter Order or FIR Filters y[n] a k b k k k M N Consider two dierent values or N N=3 H(e j2πλ N=7 ) H(e j2πλ ) Higher N leads to steeper ilter response We reer to N as the order o the ilter λ - λ M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 9

20 FIR Filter Design in Matlab x[n] H(e j2πλ ) y[n] a k K b k M k N k Lowpass, highpass, and bandpass ilters can be realized by appropriately scaling the relative value o the b coeicients Higher order (i.e., higher N) leads to steeper responses Perorm FIR ilter design using ir command Frequency response observed with reqz command See Prelab portion o Lab 3 or details M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 2

21 Summary Filters can generally be classiied according to Lowpass, highpass, bandpass operation Bandwidth and order o ilter Given a cosine input to a ilter, output is: Scaled in amplitude by magnitude o ilter requency response Shited in phase by phase o ilter requency response Matlab operates on discrete-time signals Use DTFT or analytical analysis Use commands such as ir, reqz, and ilter or design and implementation o FIR ilters Next lecture: introduce I/Q modulation and urther discuss continuous-time iltering M.H. Perrott 27 Filtering in Continuous and Discrete Time, Slide 2

AM Receiver. Prelab. baseband

AM Receiver. Prelab. baseband AM Receiver Prelab In this experiment you will use what you learned in your previous lab sessions to make an AM receiver circuit. You will construct an envelope detector AM receiver. P1) Introduction One