Digital Signal Processing (dsp) Fundamentals
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Digital Signal Processing (DSP) Fundamentals
Overview • What is DSP? • Converting Analog into Digital – Electronically – Computationally
• How Does It Work? – Faithful Duplication – Resolution Trade-offs
What is DSP? • Converting a continuously changing waveform (analog) into a series of discrete levels (digital)
What is DSP? • The analog waveform is sliced into equal segments and the waveform amplitude is measured in the middle of each segment • The collection of measurements make up the digital representation of the waveform
-1
-1.5
-2 17
15
13
11
9
7
5
0
0
0.5
3
1
0.22 0.44 0.64 0.82 0.98 1.11 1.2 1.24 1.27 1.24 1.2 1.11 0.98 0.82 0.64 0.44 0.22
1.5
19 -0.22 -0.44 21 -0.64 -0.82 23 -0.98 -1.11 25 -1.2 -1.26 27 -1.28 -1.26 29 -1.2 -1.11 31 -0.98 -0.82 33 -0.64 -0.44 35 -0.22 37
-0.5 0
0 1
What is DSP?
2
Converting Analog into Digital Electronically
• The device that does the conversion is called an Analog to Digital Converter (ADC) • There is a device that converts digital to analog that is called a Digital to Analog Converter (DAC)
Converting Analog into Digital Electronically
• The simplest form of ADC uses a resistance ladder to switch in the appropriate number of resistors in series to create the desired voltage that is compared to the input (unknown) voltage
SW-8 V-high SW-7 V-7 SW-6 V-6 SW-5 Output
V-5 SW-4 V-4 SW-3 V-3 SW-2 V-2 SW-1 V-1
V-low
Converting Analog into Digital Electronically
• The output of the resistance ladder is compared to the Analog Voltage analog voltage in a Resistance comparator Ladder Voltage • When there is a match, the digital equivalent (switch configuration) is captured
Comparator Output
Higher Equal Lower
Converting Analog into Digital Computationally
• The analog voltage can now be compared with the digitally generated voltage in the comparator • Through a technique called binary search, the digitally generated voltage is adjusted in steps until it is equal (within tolerances) to the analog voltage • When the two are equal, the digital value of the voltage is the outcome
Converting Analog into Digital Computationally
• The binary search is a mathematical technique that uses an initial guess, the expected high, and the expected low in a simple computation to refine a new guess • The computation continues until the refined guess matches the actual value (or until the maximum number of calculations is reached) • The following sequence takes you through a binary search computation
Binary Search Analog
• Initial conditions – – – –
Expected high 5-volts Expected low 0-volts 5-volts 256-binary 0-volts 0-binary
5-volts 3.42-volts
2.5-volts
Digital 256 Unknown (175) 128
• Voltage to be converted – 3.42-volts – Equates to 175 binary
0-volts
0
Binary Search • Binary search algorithm: High Low Low NewGuess 2
• First Guess:
Analog 5-volts
Digital 256
3.42-volts
unknown 128
256 0 0 128 2 Guess is Low
0-volts
0
Binary Search • New Guess (2):
256 128 128 192 2
Analog 5-volts
3.42-volts
Guess is High 0-volts
Digital 256 192 unknown
0
Binary Search • New Guess (3):
192 128 128 160 2
Analog 5-volts
Digital 256
3.42-volts
unknown 160
Guess is Low 0-volts
0
Binary Search • New Guess (4):
192 160 160 176 2
Analog 5-volts
Digital 256
3.42-volts
176 unknown
Guess is High 0-volts
0
Binary Search • New Guess (5):
176 160 160 168 2 Guess is Low
Analog 5-volts
Digital 256
3.42-volts
unknown 168
0-volts
0
Binary Search • New Guess (6):
176 168 168 172 2
Analog 5-volts
Digital 256
3.42-volts
unknown 172
Guess is Low 0-volts
(but getting close)
0
Binary Search • New Guess (7):
Analog 5-volts
3.42-volts 176 172 172 174 2
Digital 256
unknown 174
Guess is Low (but getting really, really, close)
0-volts
0
Binary Search • New Guess (8):
176 174 174 175 2
Analog 5-volts
3.42-volts
Digital 256
175!
Guess is Right On 0-volts
0
Binary Search • The speed the binary search is accomplished depends on: – The clock speed of the ADC – The number of bits resolution – Can be shortened by a good guess (but usually is not worth the effort)
How Does It Work? Faithful Duplication
• Now that we can slice up a waveform and convert it into digital form, let’s take a look at how it is used in DSP • Draw a simple waveform on graph paper – Scale appropriately
• “Gather” digital data points to represent the waveform
Starting Waveform Used to Create Digital Data
How Does It Work? Faithful Duplication
• Swap your waveform data with a partner • Using the data, recreate the waveform on a sheet of graph paper
Waveform Created from Digital Data
How Does It Work? Faithful Duplication
• Compare the original with the recreating, note similarities and differences
How Does It Work? Faithful Duplication
• Once the waveform is in digital form, the real power of DSP can be realized by mathematical manipulation of the data • Using EXCEL spreadsheet software can assist in manipulating the data and making graphs quickly • Let’s first do a little filtering of noise
How Does It Work? Faithful Duplication
• Using your raw digital data, create a new table of data that averages three data points – Average the point before and the point after with the point in the middle – Enter all data in EXCEL to help with graphing
Noise Filtering Using Averaging Ave before/after
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150
Time
10
20
Time
30
40
How Does It Work? Faithful Duplication
• Let’s take care of some static crashes that cause some interference • Using your raw digital data, create a new table of data that replaces extreme high and low values: – Replace values greater than 100 with 100 – Replace values less than -100 with -100
Clipping of Static Crashes eliminate extremes (100/-100)
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Now let’s take a look at how sampling rates affect the faithful duplication of the waveform • Using your raw digital data, create a new table of data and delete every other data point • This is the same as sampling at half the rate
Half Sample Rate every 2nd
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete every second and third data point • This is the same as sampling at one-third the rate
1/2 Sample Rate every 3rd
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete all but every sixth data point • This is the same as sampling at one-sixth the rate
1/6 Sample Rate every 6th
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete all but every twelfth data point • This is the same as sampling at one-twelfth the rate
1/12 Sample Rate every 12th
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150
Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• What conclusions can you draw from the changes in sampling rate? • At what point does the waveform get too corrupted by the reduced number of samples? • Is there a point where more samples does not appear to improve the quality of the duplication?
How Does It Work? Resolution Trade-offs Bit Resolution
Sample Rate
High Bit Count
Good Duplication
Slow
Low Bit Count
Poor Duplication
Fast
High Sample Rate
Good Duplication
Slow
Low Sample Rate
Poor Duplication
Fast
Overview • What is DSP? • Converting Analog into Digital – Electronically – Computationally
• How Does It Work? – Faithful Duplication – Resolution Trade-offs
What is DSP? • Converting a continuously changing waveform (analog) into a series of discrete levels (digital)
What is DSP? • The analog waveform is sliced into equal segments and the waveform amplitude is measured in the middle of each segment • The collection of measurements make up the digital representation of the waveform
-1
-1.5
-2 17
15
13
11
9
7
5
0
0
0.5
3
1
0.22 0.44 0.64 0.82 0.98 1.11 1.2 1.24 1.27 1.24 1.2 1.11 0.98 0.82 0.64 0.44 0.22
1.5
19 -0.22 -0.44 21 -0.64 -0.82 23 -0.98 -1.11 25 -1.2 -1.26 27 -1.28 -1.26 29 -1.2 -1.11 31 -0.98 -0.82 33 -0.64 -0.44 35 -0.22 37
-0.5 0
0 1
What is DSP?
2
Converting Analog into Digital Electronically
• The device that does the conversion is called an Analog to Digital Converter (ADC) • There is a device that converts digital to analog that is called a Digital to Analog Converter (DAC)
Converting Analog into Digital Electronically
• The simplest form of ADC uses a resistance ladder to switch in the appropriate number of resistors in series to create the desired voltage that is compared to the input (unknown) voltage
SW-8 V-high SW-7 V-7 SW-6 V-6 SW-5 Output
V-5 SW-4 V-4 SW-3 V-3 SW-2 V-2 SW-1 V-1
V-low
Converting Analog into Digital Electronically
• The output of the resistance ladder is compared to the Analog Voltage analog voltage in a Resistance comparator Ladder Voltage • When there is a match, the digital equivalent (switch configuration) is captured
Comparator Output
Higher Equal Lower
Converting Analog into Digital Computationally
• The analog voltage can now be compared with the digitally generated voltage in the comparator • Through a technique called binary search, the digitally generated voltage is adjusted in steps until it is equal (within tolerances) to the analog voltage • When the two are equal, the digital value of the voltage is the outcome
Converting Analog into Digital Computationally
• The binary search is a mathematical technique that uses an initial guess, the expected high, and the expected low in a simple computation to refine a new guess • The computation continues until the refined guess matches the actual value (or until the maximum number of calculations is reached) • The following sequence takes you through a binary search computation
Binary Search Analog
• Initial conditions – – – –
Expected high 5-volts Expected low 0-volts 5-volts 256-binary 0-volts 0-binary
5-volts 3.42-volts
2.5-volts
Digital 256 Unknown (175) 128
• Voltage to be converted – 3.42-volts – Equates to 175 binary
0-volts
0
Binary Search • Binary search algorithm: High Low Low NewGuess 2
• First Guess:
Analog 5-volts
Digital 256
3.42-volts
unknown 128
256 0 0 128 2 Guess is Low
0-volts
0
Binary Search • New Guess (2):
256 128 128 192 2
Analog 5-volts
3.42-volts
Guess is High 0-volts
Digital 256 192 unknown
0
Binary Search • New Guess (3):
192 128 128 160 2
Analog 5-volts
Digital 256
3.42-volts
unknown 160
Guess is Low 0-volts
0
Binary Search • New Guess (4):
192 160 160 176 2
Analog 5-volts
Digital 256
3.42-volts
176 unknown
Guess is High 0-volts
0
Binary Search • New Guess (5):
176 160 160 168 2 Guess is Low
Analog 5-volts
Digital 256
3.42-volts
unknown 168
0-volts
0
Binary Search • New Guess (6):
176 168 168 172 2
Analog 5-volts
Digital 256
3.42-volts
unknown 172
Guess is Low 0-volts
(but getting close)
0
Binary Search • New Guess (7):
Analog 5-volts
3.42-volts 176 172 172 174 2
Digital 256
unknown 174
Guess is Low (but getting really, really, close)
0-volts
0
Binary Search • New Guess (8):
176 174 174 175 2
Analog 5-volts
3.42-volts
Digital 256
175!
Guess is Right On 0-volts
0
Binary Search • The speed the binary search is accomplished depends on: – The clock speed of the ADC – The number of bits resolution – Can be shortened by a good guess (but usually is not worth the effort)
How Does It Work? Faithful Duplication
• Now that we can slice up a waveform and convert it into digital form, let’s take a look at how it is used in DSP • Draw a simple waveform on graph paper – Scale appropriately
• “Gather” digital data points to represent the waveform
Starting Waveform Used to Create Digital Data
How Does It Work? Faithful Duplication
• Swap your waveform data with a partner • Using the data, recreate the waveform on a sheet of graph paper
Waveform Created from Digital Data
How Does It Work? Faithful Duplication
• Compare the original with the recreating, note similarities and differences
How Does It Work? Faithful Duplication
• Once the waveform is in digital form, the real power of DSP can be realized by mathematical manipulation of the data • Using EXCEL spreadsheet software can assist in manipulating the data and making graphs quickly • Let’s first do a little filtering of noise
How Does It Work? Faithful Duplication
• Using your raw digital data, create a new table of data that averages three data points – Average the point before and the point after with the point in the middle – Enter all data in EXCEL to help with graphing
Noise Filtering Using Averaging Ave before/after
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150
Time
10
20
Time
30
40
How Does It Work? Faithful Duplication
• Let’s take care of some static crashes that cause some interference • Using your raw digital data, create a new table of data that replaces extreme high and low values: – Replace values greater than 100 with 100 – Replace values less than -100 with -100
Clipping of Static Crashes eliminate extremes (100/-100)
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Now let’s take a look at how sampling rates affect the faithful duplication of the waveform • Using your raw digital data, create a new table of data and delete every other data point • This is the same as sampling at half the rate
Half Sample Rate every 2nd
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete every second and third data point • This is the same as sampling at one-third the rate
1/2 Sample Rate every 3rd
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete all but every sixth data point • This is the same as sampling at one-sixth the rate
1/6 Sample Rate every 6th
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150 Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• Using your raw digital data, create a new table of data and delete all but every twelfth data point • This is the same as sampling at one-twelfth the rate
1/12 Sample Rate every 12th
150
150
100
100
50 0 -50 0
10
20
30
40
Amplitude
Amplitude
Raw
50 0 -50 0
-100
-100
-150
-150
Time
10
20
Time
30
40
How Does It Work? Resolution Trade-offs
• What conclusions can you draw from the changes in sampling rate? • At what point does the waveform get too corrupted by the reduced number of samples? • Is there a point where more samples does not appear to improve the quality of the duplication?
How Does It Work? Resolution Trade-offs Bit Resolution
Sample Rate
High Bit Count
Good Duplication
Slow
Low Bit Count
Poor Duplication
Fast
High Sample Rate
Good Duplication
Slow
Low Sample Rate
Poor Duplication
Fast
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