Signals, Signal Characterization, and Sampling

Deterministic Signals

  Deterministic Signals are signals which are completely specified as a function of time.

Examples of deterministic signals:  

1.  
2.   
3. (unit step function)    
4. (impulse function)    
5. sinc(t)=  
6. (``Rectangle function'').
7. (``Triangle function'').
8. Sgn(x) (``Signum function'')
9. (``Impulse train'')

Random Signals

  Random Signals are signals that take on a random value at any given instant of time. These signals must be modeled probabilistically.

Energy Signals

  The normalized energy E of the signal is

When , is classified as an energy signal.

Examples: and are energy signals. Periodic functions are not energy signals.  

Power Signals

  The normalized power P of the signal is

When , is classified as a power signal. Periodic signals are an important subgroup of power signals. If is periodic with period , then

Examples: Periodic functions such as are power signals.    Examples of signals which are neither power nor energy signals include and for t > 0.

Time Averages

  The time average of the signal is denoted by angle brackets , i.e.,

If is periodic with period , then

is the DC value of .

is the mean-square value of . This is the normalized power of .

is the root-mean-square (rms) value of .

Autocorrelation

Energy Signals: The autocorrelation of the energy signal is

The Fourier Transform of the autocorrelation is

Rayleigh's Energy Theorem: Let be the energy spectral density of the energy signal . Then

Power Signals: The autocorrelation of the power signal is

Wiener-Kinchine Theorem: Let be the power spectral density of the power signal . Then

so that

with equality holding only in the limit.

Periodic Signals: Periodic signals are a special case of power signals where the autocorrelation takes on a special form. If is periodic with period , then it may be expressed as

Then the autocorrelation is

The power spectral density of is

and the power P is

The last step is the result of Parseval's Theorem which states that

Linear Systems: Let be the input to an LTI system with impulse response and let be the resulting output. Then

and

Example: Homework problem 0.2

Bandwidth

The concept of power spectral density permits a useful definition of bandwidth. For engineering terms, the bandwidth is measured on the positive frequency axis only. The bandwidth is given by for bandpass signals. For baseband signals, . The frequencies and are determined using several common definitions:

1. Absolute Bandwidth
   and are defined following the condition

   

2. 3-dB Bandwidth
   and are determined by the range of frequencies for which

   

   where is the frequency for which is the maximum.

3. Equivalent Noise Bandwidth
   is the width of a fictitious rectangular filter such that the power in that rectangular band is equal to the actual power of the signal. The actual power of the signal is

   

   The equivalent in the rectangular band is

   

   where is the frequency for which is a maximum. Equating the actual and equivalent powers yields the relationship

   

4. Null-to-Null Bandwidth
   Let be the frequency for which is a maximum.
   1. Baseband Signals

      

   2. Bandpass Signals

      

   Note that this definition does not apply to all spectra since not all spectra have nulls.
5. Bounded Spectrum Bandwidth
   and are defined by the range of f for which

   

   Note that is usually expressed in dB.

6. Power Bandwidth
   and are defined by the frequency interval which contains some percentage R of the total signal power:

   

   Common values of R include

   1. 
   2. 
7. Spectral Mask (FCC Regulations)
   Section 12.106 of the FCC Rules and Regulations states

   ``For operating frequencies below 15 GHz, in any 4 kHz band, the center frequency of which is removed from the assigned frequency by more than 50 percent up to and including 250 percent of the authorized bandwidth, as specified by the following equation, but in no event less than 50 dB:

   

   (attenuation greater than 80 dB is not required)
   where

   

Ideal Sampling

Sampling Theorem: Let be a baseband signal which is strictly bandlimited to W Hz. (i.e. for |f| > W). is completely described by uniformly spaced (in time) instantaneous samples with period . The lowest possible sample rate is called the Nyquist frequency.

In other words, a bandlimited signal can be completely represented by its sample values.

Let be the sampled version of . Ideal instantaneous sampling may be represented by

Signal Reconstruction If the conditions of the sampling theorem are satisfied, the recovery of from is possible by using a low pass filter.

(illustration)

If the conditions of the sampling theorem are not satisfied (either because or is not bandlimited), then distortion is present at the output of the reconstruction filter:

1. so that the replicated spectra overlap. This is called aliasing.
2. The reconstruction filter does not have sufficiently high rolloff. This distortion can be reduced by increasing the filter order or increasing .

Pulse Code Modulation (PCM)

In PCM, the signal samples are quantized to discrete levels. The quantization level of each sample is transmitted instead of the sample value.

Hilbert Transform

The Hilbert Transform of the signal is defined to be the signal whose frequency components are all phase shifted by radians. The resulting signal is denoted

is produced by passing through a filter with transfer function

 

The magnitude and phase of are

The impulse response is the inverse Fourier transform of ():

It is instructive to contrast and compare the transfer function of the Hilbert transmform to that of a pure time delay (). The transfer function of the time delay is

Both have the same magnitude but the time delay has a phase which is linear in frequency instead of constant.

Example 1: Find when .

Example 2: Homework problem.

The properties of the Hilbert transform are outlined in Section 2.9 of the Text.

(Note: the Hilbert transform is used in complex analysis to generate complex-valued analytic functions from real functions. A function is analytic if and only if its components are harmonic conjugates. The Hilbert transform is used to generate a functions whose components are harmonic conjugates.)