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.