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Continuous Time Signals. A signal represents the evolution of a physical quantity in time. Example: the electric signal out of a microphone. At every time t the signal has a value Volts (say). Digital Processing of Continuous Time Signals. ADC. DSP. DAC. - PowerPoint PPT Presentation
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Continuous Time Signals
A signal represents the evolution of a physical quantity in time.
Example: the electric signal out of a microphone.
t
)(tx
t
At every time t the signal has a value Volts (say) )(tx
Digital Processing of Continuous Time Signals
ADC DSP
DAC
Signals can be processed numerically by a digital computer or using a DSP chip. We need:
1. Analog to Digital Converter (ADC): convert the signal to a numerical sequence
2. Digital to Analog Converter (DAC): convert it back to analog, if we need to.
Analog to Digital Converter (ADC)
It performs Sampling and Quantization.
sFsT
Sampling frequency (Hz=1/sec)
Sampling interval (sec)
Parameters:
BN Number of Bits per Sample
)(tx )(][ snTxQnx
sF
ADC
sT000
001
011
101
110
Digital to Analog Converter (DAC)
][nx
sF
DAC
)(txt
t
It converts a signal back to Continuous Time by holding the value within the sampling interval.
Energy of a Signal
A signal represents a physical quantity, like a Voltage, a Current, a Pressure …
We define its total Energy as:
dttxEX
2)(
Example:
sec1050100.2)0.5( 2332 VoltsEX
)(tx
sec)(mt
V0.5
0.2
Power of a Signal
A signal represents a physical quantity, like a Voltage, a Current, a Pressure …
We define the Average Power:
2/
2/
2)(
1lim
T
TT
X dttxT
P
In particular if the signal is a periodic repetition of a pulse:
period 0 T
period
2
0
)(1
dttxT
PX
Example
Take a square wave. Suppose it is a voltage and the values are in Volts:
sec)(mt
)(tx
0 5.1 0.3
5.0
23
32
Volts 125.0103
105.15.0
XP
Its square root is called the Root Mean Square (RMS) value:
Volts 35.0 XRMS PX
Relative Power: deciBells (dBs)
In many problems we are interested in the relative power, with respect to the power of a reference signal.
For example, suppose the reference has a power
Then, in the previous example:
dBP
PP
Xref
XdBX 97.10
01.0
125.0log10log10 1010
2Volts 01.0XrefP
You could use the RMS values and obtain the same result:
RMS
RMS
Xref
XdBX Xref
X
P
PP 1010 log20log10
Some Typical Values for Acoustic Signals
Take the air pressure of an audio signal. Let the reference be the threshold of hearing. For a typical person this
212 /10 mWattsPref dBP
dBref 010
10log10
12
12
10
Threshold
ppp
p
f
fff
pain
2/mWatts dB
1210
810
610
410
210
1
0
40
60
80
100
120
Signal to Noise Ratio
Usually all the signals we don’t want we call them “noise”. This can be caused by actual background noise, interference from another source (someone talking during the movie) or any other undesired sources.
signal
noise
what we get
)(tx
)(tw
)(ty
The Signal to Noise Ratio (SNR) characterizes how “noisy” the signal is:
dBWdBXW
X PPP
PSNR 10log10
Example
1. You hearing something at a level “f” (forte), and someone talks at level “ppp” (pianissimo), then the SNR is (refer to the table):
dBSNR 404080
2. You hearing something at a level “f” (forte), and someone talks at level “p” (piano), then the SNR is (refer to the table):
dBSNR 206080
Quantization Noise
Back to Discrete Time Signals. When we quantize a signal with a finite number of bits, we introduce errors which are perceived as noise.
Problem: what is the relation between number of bits per sample and SNR?
For an average signal, statistically it can be shown that: signalNerror PPB223
1
sampleper bitsBN
Then
dBNSNR BNB 02.677.423log10 2
10
original
sT000
001
011
101
110
quantized
error
t
Example
We want to determine the number of bits per sample to obtain a good SNR of at least 100dB.
Then we need:
10002.677.4 BN
which yields
8.1502.6/23.95 BN
Then we need at least 16 bits per sample.