ee421, fall 1998 michigan technological university timothy j. schulz 08-sept, 98ee421, lecture 11...
TRANSCRIPT
08-Sept, 98 EE421, Lecture 1 1
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Digital Signal Processing (DSP) Systems
Digital processing of analog signals (mixed signal applications) forms one of the most important applications of DSP theory.
A/Dconverter
DSPsystem
D/Aconverter
...101011... …001010...
analog input
digital input
analog output
digital output
antialiasingprefilter
samplingand
quantization
discreteto
continuous
reconstructionfilter
08-Sept, 98 EE421, Lecture 1 2
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Spectral Representation of Continuous-Time Signals
Fourier Analysis:
– f represents frequency in units of cycles/second or Hz and represents frequency in units of radians/second.
– If x(t) is a voltage signal, then X( and X(f) have units of volts/Hz.
– The conversion between frequency variables is = 2f.
Fourier Synthesis:
dtetxX tj
)()( dtetxfX ft2j
)()(or
2d
eXtx tj
)()( or dfefXtx ft2j
)()(
08-Sept, 98 EE421, Lecture 1 3
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Some Important Fourier Pairs
Constant (DC) signal: (this signal contains only DC or zero
frequency)
Impulse: (this signal contains all frequencies)
Complex exponential (sinusoid): (this signal contains only one
frequency component - in fact, this signal is used to define frequency!)
Real sinusoid: (this signal contains two frequency components, +/- f0)
These pairs are for frequency measured in Hz. Remember the following rule for changing variables with impulses:
)( fAA
1t )(
)( 0tf2j ffe 0
)()()cos( 0j
0j
0 ffe21
ffe21
tf2
)()(
2f2
f
08-Sept, 98 EE421, Lecture 1 4
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Signals are “Sums of Sinusoids”
Periodic signals contain only discrete frequency components that are multiples of the fundamental frequency.
Non-periodic signals contain a continuous set of frequency components.
The amplitude and phase of eachsinusoid forms the spectrum of the signal!
08-Sept, 98 EE421, Lecture 1 5
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Linear, Time-Invariant (LTI) Systems
Impulse response:
Convolution:
Frequency Response:
t t0 0
)(t)(th
)(tx
dxthty )()()(
)cos( ft2 )(cos)( fHft2fH
)()( fHth
08-Sept, 98 EE421, Lecture 1 6
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Filtering
H(f) modifies the amplitude of the input signal’s spectrum according to :
H(f) modifies the phase of the input signal’s spectrum according to:
)( fX )()()( fXfHfY
)()()( fXfHfY
)()()( fHfXfY
08-Sept, 98 EE421, Lecture 1 7
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Review of Sampling
A bandlimited signal is one whose frequency spectrum contains no components greater than some maximum frequency fmax.
The sampling theorem states that bandlimited signals can be reconstructed perfectly from their samples provided the sampling rate fs (in samples/second) satisfies:
2 fmax is called the Nyquist rate.
0 fmax-fmax f f0
maxf2fs
X(f) X(f)
08-Sept, 98 EE421, Lecture 1 8
EE421, Fall 1998Michigan Technological University
Timothy J. Schulz
Aliasing
Sampling at a rate slower than the Nyquist rate will result in aliasing. That is, frequency components greater than fs / 2 will be folded back into the Nyquist interval. This is generally a bad thing.
0 fs-fsf (Hz)
0 fs-fsf (Hz)
Don’t let this happen to you!