sampling, reconstruction, and elementary digital filters
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Sampling, Reconstruction, and Elementary Digital Filters. R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2002. Sampling and Reconstruction. - PowerPoint PPT PresentationTRANSCRIPT
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Sampling, Reconstruction, and Elementary Digital Filters
R.C. Maher
ECEN4002/5002 DSP Laboratory
Spring 2002
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 2
Sampling and Reconstruction
• Need to understand relationship between a continuous-time signal f(t) and a discrete-time (sampled) signal f(kT), where T is the time between samples (T=1/fs)
dejFkTf kTj)(2
1)(
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 3
Sampling (cont.)
• After some manipulation, can show:
n
T
T
kTj
n
T
njF
TDTFT
deT
njF
T
TkTf
21
21
2)(
1
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 4
Sampling Effects: Frequency DomainXc(j)
N-N
XS(j)
N-N S-S 2S-2S
S-S 2S-2S
XS(j)
S > 2 N
S < 2 N (aliasing)
Fourier Transform of continuous function
Fourier Transform of sampled function
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 5
Reconstruction
• Since spectrum of sampled signal consists of baseband spectrum and spectral images shifted at multiples of 2π/T, reconstruction means isolating the baseband image
• Concept: lowpass filter to pass baseband while removing images
XS(j)
N-N S-S 2S-2S
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 6
Reconstruction (cont.)
• Multiplication by rectangular pulse in frequency domain (LPF) corresponds to convolution by sinc( ) function in time domain
• Because sinc( ) is non-causal and of infinite extent, practical reconstruction requires an approximation to the ideal case
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 7
Delay Lines
• In order to create a frequency-selective function, there must be a delay memory so that the function is able to observe and resolve the frequencies present in the signal
• Digital filters used tapped delay lines to create the z-1 (delay) terms in the z-transform
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 8
Delay Lines (cont.)
Z-1
Z-1
Z-1
+h0
h1
h2
h3
x[n]
x[n-1]
y[n]
x[n-2]
x[n-3]
3
0
)(n
nn zhzH
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 9
Delay Lines (cont.)
• Delay lines can be implemented easily as a one dimensional array or FIFO in DSP memory
• Typically use an address register to point to array, then just increment pointer instead of copying data to achieve the delay
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 10
Modulo Buffers
• DSP supports modulo arithmetic in the address generation unit
• With modulo buffer, incrementing or decrementing address register “rolls over” automatically at beginning and ending of buffer memory range
• Modulo buffers are useful for delay stages in filters and other FIFO queue structures
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 11
Modulo Buffers (cont.)
• Modulo calculations keep address pointer within a fixed range of memory locations
Modulo NBuffer
Memory
N memory locations
Base Address
Base Address + N -1
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 12
56300 Modulo Buffers
• The M registers in the AGU select the modulo size of the buffer– M = $FFFFFF implies no modulo (regular
linear addressing)– M = 0 implies bit-reversed addressing (useful in
FFT algorithms)– M = ‘modulo’-1 implies address range
including ‘modulo’ memory locations (2modulo $7FFF)
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 13
56300 Modulo (cont.)
• The base address of the modulo buffer must be a power of 2
• The base address must either be zero, or a power of 2 that is greater than or equal to the modulo
• In other words, the base address must be 2k, where 2kmodulo, which implies k least significant bits must be zero
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 14
FIR Filter
• FIR filter coefficients are equal to the unit sample response of the filter
• Given filter specifications, we need to choose a unit sample response that is “close” to the desired response, yet within the implementation constraints (memory, computational complexity, etc.)
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 15
FIR Filter Design
• Several FIR design techniques are available• Consider the Window method:
– Determine ideal response function– If length of ideal function is too long, multiply
ideal response by a finite length window function
– Note that multiplication by window in time domain means convolution (and smearing) in the frequency domain
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 16
FIR Window Design Concept• Lowpass filter: cutoff at 0.2 fs .
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.2
0.4
0.6
0.8
1
1.2
Frequency (fraction of fs)
Am
plit
ud
e (
line
ar
sca
le)
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 17
FIR Design Concept (cont.)• Time domain response (Inverse DTFT)
-60 -40 -20 0 20 40 60-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Sample Index
Am
plitu
de
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 18
FIR Design Concept• Window function to limit response length
-60 -40 -20 0 20 40 60-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Sample Index
Am
plitu
de
Hamming window
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 19
FIR Design Concept (cont.)• Windowed and shifted (causal) result
0 5 10 15 20 25 30 35 40-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Am
plitu
de
Sample Index
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 20
FIR Design Concept• Resulting frequency response of filter
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-60
-50
-40
-30
-20
-10
0
10
Frequency (fraction of fs)
Mag
nitu
de (
dB)
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ECEN4002 Spring 2002 Delay Lines and Simple Filters R. C. Maher 21
Lab Assignment #2
• Due at START of class in two weeks• Topics:
– Sampling and reconstruction (MATLAB)– Program #1: Cycle counting– Program #2: Simple delay line– Program #3: File I/O via Debugger– Program #4: FIR filter, non-real time– Program #5: FIR filter, real time