Communication Systems Chapter 3 Signal Transmission and Filtering Dr. Le Dang Quang Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: ldquang@hcmut. CS-2016 1 Faculty of EEE HCMUT Chapter Outline 3.1 Response of LTI Systems 3.2 Signal Distortion in Transmission 3.3 Transmission Loss and Decibels 3.4 Filters and Filtering 3.5 Correlation and Spectral Density 3.6 Probability and Random Variables 3.7 Random Signals and Noise Telecomm. CS-2016 2 Faculty of EEE HCMUT 3.4 Filters and Filtering Telecomm.
CS-2016 56 Faculty of EEE HCMUT Filtering Every communication system includes one or more filters to separate the desired information signal from noise, distortion, and other information signals. Ideal filter An ideal filter has the characteristics of distortionless transmission over one or more specified frequency bands and zero response at all other frequencies. The transfer function is thus The bandwidth of the filter is B = fu− fl Telecomm. CS-2016 57 Faculty of EEE HCMUT Filtering Ideal filters are physically unrealizable, in the sense that their characteristics cannot be achieved with a finite number of elements.
For example, the impulse response of an ideal filter is a sinc-function, which is infinite long and noncausal. The filters used in practical applications must be causal, that is Telecomm. CS-2016 58 Faculty of EEE HCMUT Filtering Real filters Telecomm. CS-2016 59 Faculty of EEE HCMUT Filtering Real filters: Design issues Filter specifications and the amplitude response for realizable filters: Lowpass filter: Telecomm.
CS-2016 60 Faculty of EEE HCMUT Filtering The bandwidth is usually defined to be the 3 dB-bandwidth. The amplitude response at the edge frequencies fl and fu is then lowered by 3 dB (or 0.707 in the linear scale) compared to the maximum value of the passband. CS-2016 61 Faculty of EEE HCMUT Filtering Butterworth filter A Butterworth filter is the simplest of the standard (realizable) filter types which is also easy to analyse. The amplitude response of the nth-degree Butterworth filter is: The amplitude response is maximally flat, meaning that its first n derivatives equal to zero at f = 0.
CS-2016 62 Faculty of EEE HCMUT Filtering Third-order Butterworth LPF Telecomm. CS-2016 63 Faculty of EEE HCMUT Filtering Bode diagram of Butterworth filter Telecomm. CS-2016 64 Faculty of EEE HCMUT Filtering Other filter types If the amplitude response in the passband and stopband varies between certain maximum and minimum values, the filter is called an equiripple filter. Elliptic filters (Cauer filters) are equiripple in the passband and stopband.
These filters provide the best selectivity in the sense that their transition bands are the sharpest for a given filter specifications. Other filter types are the Chebyshev filters which are equiripple in the passband and maximally flat in the stopband (or vice versa.) The filters mentioned above are based on the approximation of an ideal amplitude response. When the aim is a good selectivity, the phase response may be poor. Bessel filters have a better phase response (linear phase and constant group delay).
In critical applications, the filter is optimized according to both the amplitude and phase response. Frequency transforms may be used to transform a prototype LPF into HPF, BPF, BSF, etc. CS-2016 65 Faculty of EEE HCMUT Filtering Filter implementation Implementation of the analog filters can be based on Passive RLC-circuits: usually used for high frequency (RF) applications because of the limited frequency range of active circuits and good behaviour of inductors at high frequencies. Active filters: are used at low frequencies (audio and video) because inductors exhibit a significant resistive component at low frequencies.
There are special implementations structures for microwave filters. CS-2016 66 Faculty of EEE HCMUT Filtering Pulse response and risetime The spectrum of a rectangular pulse has high frequency components. This applies also to other signals having sharp changes, such as the unit step function. Filtering these signals smoothes the sharp corner.
This smoothing effect has to be studied in the time domain. For example, the step response of a first order RC filter is Another example is the step response of an ideal lowpass filter: Here μ = 2B and Telecomm. CS-2016 67 Faculty of EEE HCMUT Filtering (12) This integral function is illustrated below Telecomm. CS-2016 68 Faculty of EEE HCMUT Filtering The step responses for the first order filters and for the ideal lowpass filters: The first-order filters do not have a good attenuation in high frequencies, and thus, their step responses rise fast.
For example, the step response of a higher-order Butterworth filter is closer to the step response of the ideal lowpass filter. CS-2016 69 Faculty of EEE HCMUT Filtering The risetime is a measure of the “speed” of the step response. It is usually defined in the time interval where the output signal rises from 10% to 90% of the final value. The risetimes for the first order and ideal lowpass filters are tr 0.
The following approximation can be used for the risetime of an arbitrary lowpass filter: The risetime of a steep slope at corners is faster (shorter rise time) when more high-frequency harmonics are included in the Fourier series. We loosely describe any waveform with sharp corners and short risetime as having “large bandwidth”. CS-2016 70 Faculty of EEE HCMUT Filtering Conversely, when we remove high frequency components from a waveform via filtering devices (i., good attenuation at high frequencies), we: reduce the bandwidth increase the risetime change the waveform to some extent Pulse response The pulse response is the filter response to a rectangular pulse (make distinction with impulse response!). The rectangular pulse can be constructed from two unit-step functions.
CS-2016 71 Faculty of EEE HCMUT Filtering The pulse response of the ideal lowpass filter is When Bτ > 2 the pulse response is close to the rectangular pulse. For lower values of Bτ , the pulse response becomes flat. CS-2016 72 Faculty of EEE HCMUT Filtering Required bandwidth for a pulse signal Rule of thumbs for the required bandwidth of a pulse: (1) If the shape of the pulse is desired to be preserved, then the required bandwidth is high: (2) The pulse is only detected or its amplitude is measured, then smaller bandwidth can be used: This is the condition for resolving two pulses separated by τmin or more. It is assumed here that the pulse response of the filter is close to the ideal response.
If this is not the case, the pulse contains more distortion. CS-2016 73 Faculty of EEE HCMUT 3.5 Correlation and Spectral Density Telecomm. CS-2016 74 Faculty of EEE HCMUT Correlation and Spectral Density Here we study signals using the time average and signal power (or energy). This leads to the concept of spectral density functions.
Spectral densities allow us to deal with a broader range of signal models, not necessarily Fourier transformable (e. Time average The time average of an arbitrary signal is defined: It has the properties: The average power of power signal (i., signal with finite power) is: Telecomm. CS-2016 75 Faculty of EEE HCMUT Correlation and Spectral Density Example: The time average of a sinusoid signal: The average power of a sinusoid signal: Scalar product The scalar product of the power signals v(t) and w(t) is denoted by v(t)w∗(t). It is real or complex and it measures the similarity between the two signals.
The Schwarz's inequality relates the scalar product and the signal powers: Equality holds when v(t) = aw(t). CS-2016 76 Faculty of EEE HCMUT Correlation and Spectral Density Let's calculate the power of the signal: z(t) = v(t) − aw(t) which is when the value of the scalar product is large, the power of signal difference v(t) − w(t) is small, thus the signals are similar. Correlation functions of a power signal The crosscorrelation is defined as: It is a function of the delay parameter and it has the properties: Telecomm. CS-2016 77 Faculty of EEE HCMUT Correlation and Spectral Density The crosscorrelation measures the similarity between v(t) and w(t− ) as a function of the time shift.
As a special case, we have the autocorrelation function: Autocorrelation tells us something about the time variation of the signal. If the autocorrelation is high for some value of , then the original and delayed signals (when delay is ) are similar. Properties of the autocorrelation function: The autocorrelation of real functions is real and even. The autocorrelation of periodic functions is periodic.
CS-2016 78 Faculty of EEE HCMUT Correlation and Spectral Density Autocorrelation of sum and difference signals: If v(t) and w(t) are uncorrelated that is then Superposition of average powers therefore holds for uncorrelated signals. CS-2016 79 Faculty of EEE HCMUT Correlation and Spectral Density Correlation of complex exponentials: The scalar product of two complex exponentials is Let's consider the following two signals (complex exponentials): where Cv and Cw are complex constants which determine the amplitude and phase of the signals. The crosscorrelation is These signals are uncorrelated if their frequencies are not equal. The autocorrelation is Telecomm.
CS-2016 80 Faculty of EEE HCMUT Correlation and Spectral Density Autocorrelation of sinusoidal signals The autocorrelation of the sinusoidal signal: is Its maximum value is: Autocorrelation is independent on the phase of the signal. This emphasizes the fact that the autocorrelation does not uniquely define a signal. CS-2016 81 Faculty of EEE HCMUT Correlation and Spectral Density Correlation functions of energy signals If the energy of the signal is finite, its time average is zero. In this case, the crosscorrelation and autocorrelation are defined as follows: and they have the same properties as the correlation functions of power signals.
In the above equations, the power (Pv) is replaced by the energy (Ev). For example, In this case, the crosscorrelation can be calculated by using the convolution: Telecomm. CS-2016 82 Faculty of EEE HCMUT Correlation and Spectral Density Based on the properties of Fourier transform and on Schwarz inequality, we can write: Telecomm. CS-2016 83 Faculty of EEE HCMUT Correlation and Spectral Density Input-output correlation For the LTI systems, the crosscorrelation between the input and output, as well as the autocorrelation of the output, are given by Telecomm.
CS-2016 84 Faculty of EEE HCMUT Correlation and Spectral Density Spectral density functions Spectral density function Gx(f) represents the distribution of the power or energy in the frequency domain. The area under Gx(f) equals the average power or total energy. It can be shown that the autocorrelation function and its spectral density function form a Fourier transform pair: that is For linear time-invariant system, the spectral density of the output is Telecomm. CS-2016 85 Faculty of EEE HCMUT Correlation and Spectral Density Power and energy spectrum If the Fourier transform of energy signal v(t) is V(f) then its energy spectral density is.
If the Fourier series coefficients of periodic power signal v(t) are cn = c(nf0) then This spectrum consists of impulses and the amplitude of these impulses represents the power of harmonic components. For example, in the case of sinusoidal: Telecomm. CS-2016 86 Faculty of EEE HCMUT Correlation and Spectral Density The power in a specified frequency band of signal x(t) is obtained by integrating its power spectral density over that frequency band: Example: Comb filter The impulse response and transfer function of this filter are Telecomm. CS-2016 87 Faculty of EEE HCMUT Correlation and Spectral Density The square of the amplitude response is and it is illustrated in the figure in previous slide (that’s why the name of “comb”).
The power spectral density of the output signal is: And the autocorrelation function of the output is: and the output power or energy is Telecomm. CS-2016 88 Faculty of EEE HCMUT 3.6 Probability and Random Variables Telecomm. CS-2016 89 Faculty of EEE HCMUT Probability and Random Variables So far we have considered deterministic signals whose behaviour is known for all possible times. Random signals occur in communication systems both as unwanted noise and as desired information-bearing signals.