Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
1
"Objectives" of Lecture on DSP
• Introduction to laboratory #1• The need for DSP• Resolution, Aliasing & Windowing• Introduction to FFT.
2
Why Digital Signal Processing?
• Allows fast & complex calculations, usuallywithout loss of information.
E.G.– MRIs
– Communication (TV,CD,MP3, etc.)– Signal diagnostics (machine health, “looking” for
hidden details)
3
A/D converters(Analogue/Digital)
This lecture, 3 concepts & 1 process.
•A/D resolution
•Aliasing (sampling rate)
•Windowing (leakage)
Time domain
Frequency domain
FFT
7
Mention of one other issue:Resolution
For a specific voltage range there will onlybe a certain number of "divisions" available (see later).
8
Mention of one other issue:Resolution
Δvoltage
For a specific voltage range there will onlybe a certain number of "divisions" available (see next slide).
9
Example of a A/D spec.
“12bit A/D converter with a sampling rate of 15kHzover a range of -/+ 5 volts”
Voltage increments:Δvoltage = 10/212 = 10/4096 = 2.441mVTime increment:Δt = 1/15,000 sec.
10
Aliasing
• How fast do we have to sample?• What happens if we don't sample fast
enough?• What can we do about it?
14
We must sample AT LEAST twice as fast as the highestfrequency that is present.e.g. here, 4Hz signal therefore sample 8Hz.
!
>
“Onset” of aliasing
15
• To avoid aliasing, the analogue signal isfirst FILTERED (anti-aliasing filters) toensure that all frequencies higher than 1/2the sampling frequency have been removed.
• The maximum detectable frequency issometimes called the Nyquist frequency.
SUMMARY
18
Windowing
The Fourier series algorithim believes thisfinite record, of length T.R., to be thefundamental part of a periodic function. Itattempts to recreate a function as sketchedbelow. The sudden “stop” and “start”,shown in red, causes the creation ofadditional components in the frequencydomain. These additional frequencies areknown as LEAKAGE. This can beminimized by tapering the beginning andending of each T.R. to zero.
21
Recap Fourier series.
x t( ) =1
Ta0
+ ancos !
nt( ) + b
nsin !
nt( )
n=1
"
#$ % &
' &
( ) &
* &
Periodic signal of period τ, x(t) = x(t + !)
where !n=2n"
#n = 1,2, 3,.....
an=2
!x(t)cos("
nt)dt
#! / 2
! / 2
$
bn=2
!x(t)sin("
nt)dt
#! / 2
! / 2
$
and where
See eq(1.2.1-3)
22
Fourier series and FFT
x(t) =4
!sin("1t) +
4
3!sin(3"1t) +
4
5!sin(5"1t) + ....
Recall that for the following square wave:
x(t)
t
1
23
Another way of viewing this:x(t)
t
frequency
4
!
4
3!4
5!
!12!
13!
1
Magnitude of Sine component
Time domain
Frequency domain
FFT