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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.

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slide1

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)

slide2

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:
  • Analog to Digital Converter (ADC): convert the signal to a numerical sequence
  • Digital to Analog Converter (DAC): convert it back to analog, if we need to.
slide3

Analog to Digital Converter (ADC)

ADC

It performs Sampling and Quantization.

Parameters:

Sampling interval (sec)

Sampling frequency (Hz=1/sec)

Number of Bits per Sample

slide4

Digital to Analog Converter (DAC)

DAC

It converts a signal back to Continuous Time by holding the value within the sampling interval.

slide5

Energy of a Signal

A signal represents a physical quantity, like a Voltage, a Current, a Pressure …

We define its total Energy as:

Example:

slide6

Power of a Signal

A signal represents a physical quantity, like a Voltage, a Current, a Pressure …

We define the Average Power:

In particular if the signal is a periodic repetition of a pulse:

slide7

Example

Take a square wave. Suppose it is a voltage and the values are in Volts:

Its square root is called the Root Mean Square (RMS) value:

slide8

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:

You could use the RMS values and obtain the same result:

slide9

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

slide10

Signal to Noise Ratio

noise

signal

what we get

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.

The Signal to Noise Ratio (SNR) characterizes how “noisy” the signal is:

slide11

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):

2. You hearing something at a level “f”(forte), and someone talks at level “p” (piano), then the SNR is (refer to the table):

slide12

Quantization Noise

original

quantized

error

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:

Then

slide13

Example

We want to determine the number of bits per sample to obtain a good SNR of at least 100dB.

Then we need:

which yields

Then we need at least 16 bits per sample.