Tutorial using MATLAB** By, Deborah Goshorn dgoshorn@cs.ucsd.edu - - PowerPoint PPT Presentation

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Tutorial using MATLAB** By, Deborah Goshorn dgoshorn@cs.ucsd.edu - - PowerPoint PPT Presentation

Nov 02, 2023 152 likes •396 views

Multirate Signal Processing* Tutorial using MATLAB** By, Deborah Goshorn dgoshorn@cs.ucsd.edu I. Signal processing background II. Downsample Example III. Upsample Example * Multrate signal processing is used for the practical applications in

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Multirate Signal Processing* Tutorial using MATLAB**

  • I. Signal processing background
  • II. Downsample Example
  • III. Upsample Example

* Multrate signal processing is used for the practical applications in signal processing to save costs, processing time, and many other practical reasons. ** MATLAB is an industry standard software which performed all computations and corresponding figures in this tutorial

By, Deborah Goshorn dgoshorn@cs.ucsd.edu

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SLIDE 2
  • I. Signal processing

background

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SLIDE 3

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Frequency (Hz) Signal Strength

Receive an analog signal

  • Receive an analog signal at 5 Hz

(as pictured below left, there are 5 wave cycles in one second.)

  • The highest frequency component (5 Hz) of the signal is

called the signal’s bandwidth, BW, since in the examples in this presentation, the minimum frequency component is 0Hz.

  • This signal can be represented in two ways:

time representation (sec) frequency representation (Hz)

Peak signal strength at 5 Hz

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  • 1
  • 0.5

0.5 1

Time (sec) Signal Value

BW

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SLIDE 4

Add high frequency components

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  • 1
  • 0.5

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Time (sec) Signal Value

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Frequency (Hz) Signal Strength

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  • 1

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Time (sec) Signal Value

  • 2. Add a

10 Hz component

  • 3. Then add

a 15 Hz component!

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  • 1
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Time (sec) Signal Value

  • 1. Original

5 Hz signal

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Frequency (Hz) Signal Strength

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Frequency (Hz) Signal Strength

Adding high frequency components creates jagged edges in the original 5 Hz signal. BW = 5 Hz BW = 10 Hz BW = 15 Hz

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SLIDE 5

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Frequency (Hz) Signal Strength

  • In order to sample the signal without losing

information, use a sampling rate (SR) of at least the Nyquist Rate (NR), which is 2 x BW of the received analog signal.

Signal bandwidth BW = 15 Hz Nyquist Rate NR = 2 x 15Hz = 30 Hz

RULE: Sampling Rate SR ≥ Nyquist Rate NR

Sampling the signal: Nyquist Rate

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SLIDE 6

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Frequency (Hz) Signal Strength

Since Bandwidth BW = 15 Hz, the Nyquist Rate NR = 2 x 15Hz = 30Hz.

RULE #1: Sampling Rate SR ≥ Nyquist Rate NR

Signal bandwidth BW = 15 Hz Nyquist Rate NR = 30 Hz Sample Rate SR = 40 Hz

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Time (sec) Signal Value

Let Sample Rate SR = 40 Hz, so sample signal every 0.025 sec (25 milliseconds).

Sampling the signal: Nyquist Rate

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SLIDE 7

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Frequency (Hz) Signal Strength

Sampling the signal: Nyquist Freq

  • The Nyquist Frequency (NF) is equal to half of the sampling rate

(SR). The NF must be equal to or greater than the bandwidth BW of the desired signal to reconstruct.

Signal bandwidth BW = 15 Hz Nyquist Freq NF = 40/2 = 20 Hz

Rule #2: Nyquist Frequency NF ≥ Bandwidth BW

Sample Rate SR = 40 Hz

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SLIDE 8
  • II. Downsample Example
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SLIDE 9

Recall, our original signal at 5Hz…

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  • 1

1 2 3

Time (sec) Signal Value

  • 2. We added

10 & 15 Hz components!

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  • 1
  • 0.5

0.5 1

Time (sec) Signal Value

  • 1. Original

5 Hz signal

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Frequency (Hz) Signal Strength

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Frequency (Hz) Signal Strength

BW = 5 Hz BW = 15 Hz

  • 3. Then we

sampled at SR1 = 40Hz BW = 15 Hz

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  • 1

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Time (sec) Signal Value

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Frequency (Hz) Signal Strength

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SLIDE 10

Resample the sampled signal: downsampling 4

Downsample by 4 means to retain only every 4th sample

Sample Rate 1 SR1 = 40Hz Sample Rate 2 SR2 = 10 Hz NF1 = 20Hz > 15Hz = BW NF2 = 5Hz < 15Hz = BW GOOD! BAD!

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  • 1

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Time (sec) Signal Value

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  • 1

1 2 3

Time (sec) Signal Value

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SLIDE 11

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Frequency (Hz) Signal Strength

Nyquist Freq < Bandwidth 

Signal bandwidth BW = 15 Hz Nyquist Freq 2 NF2 = 10/2 = 5 Hz

Cannot recover original signal bandwidth, since new Nyquist Frequency (5Hz) is less than the desired signal bandwdidth BW (15Hz). Is the original 5Hz signal recoverable? It should be, since NF2 ≥ BW 5 Hz

Nyquist Freq 1 NF1 = 40/2 = 20 Hz

NF2 < BW means we cannot recover 15Hz BW signal

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SLIDE 12

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Frequency (Hz) Signal Strength

Why 5Hz signal not recoverable:

High Frequency band causes aliasing when downsampled

Signal bandwidth BW = 15 Hz Nyquist Freq 2 NF2 = 10/2 = 5 Hz Nyquist Freq 1 NF1 = 40/2 = 20 Hz

NF2 NF1

High frequency band

Will wrap down to 0Hz High frequency band will wrap down to 0Hz when downsampled

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SLIDE 13

Why 5Hz signal not recoverable:

Aliasing Effects

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  • 1

1 2 3

Time (sec) Signal Value

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Frequency (Hz) Signal Strength

Aliasing effects: high frequency components wrapped around to 0Hz!

Recovered 5Hz component

SR2 = 10 Hz

Due to the high frequency components at 10Hz and 15Hz that show up at 0Hz when the signal is downsampled, the 5Hz component is not recoverable. … unless we remove the high frequency components before downsampling.

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SLIDE 14

How to Remove the High Frequency components before downsampling using a low-pass filter

  • A low-pass filter (LPF) removes high frequency

components by only letting low frequency components pass through.

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Frequency (Hz) Signal Strength

It removes the jagged edges that were due to

high frequencies.

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  • 1

1 2 3

Time (sec) Signal Value

LPF LPF

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  • 1
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Time (sec) Signal Value

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Frequency (Hz) Signal Strength

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SLIDE 15

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Frequency (Hz) Signal Strength

Proof in the pudding: No more aliasing

effects when using low pass filter!

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Frequency (Hz) Signal Strength

LPF

4

SR1 = 40 Hz SR2 = 10 Hz

The original 5Hz signal is successfully recovered!

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SLIDE 16

Proof in the pudding: LPF+downsampling <==> multirate polyphase filter resampling

LPF

4

MATLAB’S* Polyphase-filter Implemented Resample (by 1/4) Function

Sample Rate 1 SR1 = 40 Hz Sample Rate 2 SR2 = 10 Hz Sample Rate 2 SR2 = 10 Hz

* MATLAB is an industry standard software which performed all computations and corresponding figures in this presentation

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Frequency (Hz) Signal Strength 5 10 15 20 25 30 35 40 1 2 3 4 5

Frequency (Hz) Signal Strength

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SLIDE 17
  • III. Upsampling example
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SLIDE 18

Assume our original signal at 5Hz…

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  • 1
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Time (sec) Signal Value

  • 1. Original

5 Hz signal

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Frequency (Hz) Signal Strength

BW = 5 Hz

  • 2. We sample

at SR1 = 15Hz BW = 5Hz

Nyquist Rate NR = 2 x BW 5Hz = 10Hz, so sample at sampling rate SR = 15Hz

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Time (sec) Signal Value

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Frequency (Hz) Signal Strength

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SLIDE 19

Resample the sampled signal: upsampling

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  • 1
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Time (sec) Signal Value

If you need to increase the number of samples in a given time by a factor of 5, you upsample by 5 (insert 5-1=4 zeros between each sample).

5

Sample Rate 1 SR1 = 15 Hz Sample Rate 2 SR2 = 75 Hz

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Time (sec) Signal Value

0.2 sec 0.2 sec

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SLIDE 20

Upsampled signal in frequency representation

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  • 1
  • 0.5

0.5 1

Time (sec) Signal Value

5

Sample Rate 1 SR1 = 15 Hz Sample Rate 2 SR2 = 75 Hz

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  • 1
  • 0.5

0.5 1

Time (sec) Signal Value

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Frequency (Hz) Signal Strength

0.2 sec 0.2 sec

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Frequency (Hz) Signal Strength

Oops!

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SLIDE 21

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Frequency (Hz) Signal Strength

Upsampling causes aliasing in higher frequencies 5

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Frequency (Hz) Signal Strength

Signal bandwidth BW = 5 Hz Signal bandwidth BW = 5 Hz Mirror Images at: 15 – 5 = 10 Hz 15 + 5 = 20 Hz 2*15 - 5 = 25 Hz 2*15 + 15 = 35 Hz … Sample Rate 1 SR1 = 15 Hz

Upsampling causes copies of the original 5Hz component at multiples

  • f original sampling rate, 15Hz, plus/minus 5Hz

How do we remove these extra high frequency components?

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SLIDE 22

How to remove the extra high frequency components caused by upsampling using a low-pass filter

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Frequency (Hz) Signal Strength

LPF

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Frequency (Hz) Signal Strength

Low pass filter removes these extra high frequency components

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SLIDE 23

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Frequency (Hz) Signal Strength

Proof in the pudding: No more aliasing

effects when using low pass filter!

SR1 = 40 Hz SR2 = 10 Hz

All high frequency copies of the 5Hz signal are removed!

LPF

5

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Frequency (Hz) Signal Strength

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SLIDE 24

5 10 15 20 25 30 35 40 50 100 150 200 250

Frequency (Hz) Signal Strength

5 10 15 20 25 30 35 40 2 4 6 8

Frequency (Hz) Signal Strength

Proof in the pudding: upsampling and lowpass filter <==> multirate polyphase filter resampling

LPF

5

MATLAB’S* Polyphase-filter Implemented Resample (by 5) Function

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Frequency (Hz) Signal Strength

Sample Rate 1 SR1 = 15 Hz Sample Rate 2 SR2 = 75 Hz Sample Rate 2 SR2 = 75 Hz

* MATLAB is an industry standard software which performed all computations and corresponding figures in this presentation