1. Eeng360 1 Chapter 2 Linear Systems Topics:  Review of Linear Systems Linear Time-Invariant Systems Impulse Response Transfer Functions Distortionless Transmission Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

2. Eeng360 2 Linear Time-Invariant Systems An electronic filter or system is Linear when Superposition holds, that is when, Where y(t) is the output and x(t) = a 1 x 1 (t)+a 2 x 2 (t) is the input. l[.] denotes the linear (differential equation) system operator acting on [.]. If the system is time invariant for any delayed input x(t – t 0 ), the output is delayed by just the same amount y(t – t 0 ). That is, the shape of the response is the same no matter when the input is applied to the system.

3. Eeng360 3 Impulse Response x(t)x(t)

4. Eeng360 4 Impulse Response The linear time-invariant system without delay blocks is described by a linear ordinary differential equation with constant coefficients and may be characterized by its impulse response h(t). The impulse response is the solution to the differential equation when the forcing function is a Dirac delta function. y(t) = h(t) when x(t) = δ(t). In physical networks, the impulse response has to be causal. h(t) = 0 for t < 0 Generally, an input waveform may be approximated by taking samples of the input at ∆t-second intervals. Then using the time-invariant and superposition properties, we can obtain the approximate output as This expression becomes the exact result as ∆t becomes zero. Letting n∆t = λ, we obtain

5. Eeng360 5 Transfer Function The output waveform for a time-invariant network can be obtained by convolving the input waveform with the impulse response of the system. The impulse response can be used to characterize the response of the system in the time domain. The spectrum of the output signal is obtained by taking the Fourier transform of both sides. Using the convolution theorem, Where H(f) = ℑ [h(t)] is transfer function or frequency response of the network. The impulse response and frequency response are a Fourier transform pair: Generally, transfer function H(f) is a complex quantity and can be written in polar form.

6. Eeng360 6 Transfer Function Transfer Function The |H(f)| is the Amplitude (or magnitude) Response. The Phase Response of the network is Since h(t) is a real function of time (for real networks), it follows 1. |H(f)| is an even function of frequency and 2. θ(f) is an odd function of frequency.

7. Eeng360 7 Output for Sinusoidal and periodic Inputs Output for Sinusoidal and periodic Inputs

8. Eeng360 8 Power Transfer Function  Derive the relationship between the power spectral density (PSD) at the input, P x (f), and that at the output, P y (f), of a linear time-invariant network. Using the definition of PSD PSD of the output is Using transfer function in a formal sense, we obtain Thus, the power transfer function of the network is

9. Eeng360 9 Example 2.14 RC Low Pass Filter

10. Eeng360 10 Example 2.14 RC Low Pass Filter 3dB Point

11. Eeng360 11 Distortionless Transmission For no distortion at the output of an LTI system, two requirements must be satisfied: 1.The amplitude response is flat. |H(f)| = Constant = A ( No Amplitude Distortion) 2.The phase response is a linear function of frequency. θ(f) = <H(f) = -2πfT d (No Phase Distortion) Second requirement is often specified equivalently by using the time delay. We define the time delay of the system as:  If T d (f) is not constant, there is phase distortion, Because the phase response θ(f), is not a linear function of frequency. No Distortion if y(t)=Ax(t-T d ) Y(f)=AX(f)e -j2  fTd

12. Eeng360 12 Example 2.15 Distortion Caused By an RC filter  For f <0.5f 0, the filter will provide almost distortionless transmission. The error in the magnitude response is less than 0.5 dB. The error in the phase is less than 2.18 (8%).  For f <f 0, The error in the magnitude response is less than 3 dB. The error in the phase is less than 12.38 (27%).  In engineering practice, this type of error is often considered to be tolerable.

13. Eeng360 13 Example 2.15 Distortion Caused By a filter  The magnitude response of the RC filter is not constant.  Distortion is introduced to the frequencies in the pass band below frequency f o.

14. Eeng360 14 Example 2.15 Distortion Caused By a filter  The phase response of the RC filter is not a linear function of frequency.  Distortion is introduced to the frequencies in the pass band below frequency f o.  The time delay across the RC filter is not constant for all frequencies.  Distortion is introduced to the frequencies in the pass band below frequency f o.