SIGNAL FILTERING AND FREQUENCY DOMAIN NOISE BASIC INFORMATION

Effect of Filtering on Signals

As we have seen, many signals have a bandwidth that is theoretically infinite. Limiting the frequency response of a channel removes some of the frequency components and causes the time-domain representation to be distorted.

An uneven frequency response will emphasize some components at the expense of others, again causing distortion. Nonlinear phase shift will also affect the time-domain representation.

For instance, shifting the phase angles of some of the frequency components in the square-wave representation changed the signal to something other than a square wave. However, while an infinite bandwidth may theoretically be required, for practical purposes quite a good representation of a square wave can be obtained with a band-limited signal.

In general, the wider the bandwidth, the better, but acceptable results can be obtained with a band-limited signal. This is welcome news, because practical systems always have finite bandwidth.

Noise in the Frequency Domain

Noise power is proportional to bandwidth. That implies that there is equal noise power in each hertz of bandwidth. Sometimes this kind of noise is called white noise, since it contains all frequencies just as white light contains all colors.

In fact, we can talk about a noise power density in watts per hertz of bandwidth. The equation for this is very simply derived. We start with Equation (1.3):

PN = kTB

This gives the total noise power in bandwidth, B. To find the power per hertz, we just divide by the bandwidth to get an even simpler equation:

N0 = kT (1.10)

where

N0 = noise power density in watts per hertz

k = Boltzmann’s constant, 1.38 × 10−23 joules/kelvin (J/K)

T = temperature in kelvins

TRANSMISSION LINE FILTERS, BALUNS AND MATCHING CIRCUITS BASIC INFORMATION AND TUTORIALS

Use can be made of standing waves on sections of line to provide filters and RF transformers. When a line one-quarter wavelength long (aλ/4 stub) is open circuit at the load end, i.e. high impedance, an effective short-circuit is presented to the source at the resonant frequency of the section of line, producing an effective band stop filter.

The same effect would be produced by a short-circuited λ/2 section. Unbalanced co-axial cables with an impedance of 50 ohm are commonly used to connect VHF and UHF base stations to their antennas although the antennas are often of a different impedance and balanced about ground.

To match the antenna to the feeder and to provide a balance to unbalance transformation (known as a balun), sections of co-axial cable are built into the antenna support boom to act as both a balun and an RF transformer.

Balun

The sleeve balun consists of an outer conducting sleeve, one quarter wavelength long at the operating frequency of the antenna, and connected to the outer conductor of the co-axial cable as in Figure 3.5.

When viewed from point Y, the outer conductor of the feeder cable and the sleeve form a short circuited quarter-wavelength stub at the operating frequency and the impedance between the two is very high.

This effectively removes the connection to ground for RF, but not for DC, of the outer conductor of the feeder cable permitting the connection of the balanced antenna to the unbalanced cable without short-circuiting one element of the antenna to ground.

RF transformer

If a transmission line is mismatched to the load variations of voltage and current, and therefore impedance, occur along its length (standing waves). If the line is of the correct length an inversion of the load impedance appears at the input end.

When a λ/4 line is terminated in other than its characteristic impedance an impedance transformation takes place. The impedance at the source is given by:

Zs = Z0^2/ ZL

where

Zs = impedance at input to line

Z0 = characteristic impedance of line

ZL = impedance of load

By inserting a quarter-wavelength section of cable having the correct characteristic impedance in a transmission line an antenna of any impedance can be matched to a standard feeder cable for a particular design frequency.

WAVE FILTERS BASIC AND TUTORIALS

Wave filter circuits are networks that contain reactive components (typically L and C) that accept or reject frequencies above or below stated cut-off frequency limits which are calculated from the values of the filter components.

Output amplitude and phase vary considerably as the signal frequency approaches a cut-off frequency and the calculations that are involved are beyond the scope of this book. The use of computer simulation, is advisable when designing such circuits.

Much more easily predictable responses can be obtained, for audio frequencies at least, by using active filters . Quartz crystals are used extensively in filter circuits to provide very sharp cut-off points, and in these applications the frequency–temperature characteristic of the devices is the most important parameter.

This leads to the use of AT-cut crystals as the preferred type, providing very good frequency stability over a wide temperature range.

Ceramic resonators, using materials such as lead zirconate titanate (PZT) are extensively used in filter circuits, and in microprocessor timing applications. These materials are piezoelectric, and can resonate in several modes depending on their resonance frequency.

Their precision of oscillation is lower than that of quartz crystals, but very much better than a discrete LC circuit, with a temperature coefficient of around 10−5/◦C in a temperature range of, typically, −10◦C to +80◦C.

They are considerably lighter and smaller than quartz crystals, and relatively immune to alterations on loading or in power supply voltage. Ceramic resonators in SM format often have load capacitors built in; other configurations may require load capacitors to be added.

TYPES AND SPECIFICATIONS OF FREQUENCY SIGNAL FILTERS BASIC ELECTRONICS INFORMATION

WHAT ARE THE TYPES AND SPECIFICATIONS OF FREQUENCY SIGNAL FILTERS?


There are many types of filter. The more popular ones are:
• Butterworth or maximally flat filter;
• Tchebyscheff (also known as Chebishev) filter;
• Cauer (or elliptical) filter for steeper attenuation slopes;
• Bessel or maximally flat group delay filter.


All of these filters have advantages and disadvantages and the one usually chosen is the filter type that suits the designer’s needs best. You should bear in mind that each of these filter types is also available in low pass, high pass, bandpass and stopband configurations.


Specifying filters
The important thing to bear in mind is that although the discussion on filters starts off by describing low pass filters, we will show you later by examples how easy it is to change a low pass filter into a high pass, a bandpass or a bandstop filter.

Figure 5.17(a) shows the transmission characteristics of an ideal low pass filter on a normalised frequency scale, i.e. the frequency variable (f) has been divided by the passband line frequency ( fp). Such an ideal filter cannot, of course, be realised in practice. For a practical filter, tolerance limits have to be imposed and it may be represented pictorially as in Figure 5.17(b).

The frequency spectrum is divided into three parts, first the passband in which the insertion loss (Ap) is to be less than a prescribed maximum loss up to a prescribed minimum frequency ( fp).

The second part is the transition limit of the passband frequency limit fp and a frequency Ws in which the transition band attenuation must be greater than its design attenuation.

The third part is the stopband limit in which the insertion loss or attenuation is to be greater than a prescribed minimum number of decibels.

Hence, the performance requirement can be specified by five parameters:
• the filter impedance Z0
• the passband maximum insertion loss (Ap)
• the passband frequency limit ( fp)
• the stopband minimum attenuation (As)
• the lower stopband frequency limit (OHMs)

Sometimes, manufacturers prefer to specify passband loss in terms of return loss ratio (RLR) or reflection coefficient (r). We provide Table 5.2 to show you the relationship between these parameters. If the values that you require are not in the table, then use the set of formulae we have provided to calculate your own values.

These parameters are inter-related by the following equations, assuming loss-less reactances: