Chapter4 Bandpass Signalling Bandpass Filtering and Linear Distortion Amplifiers and Nonlinear Distortion Total Harmonic Distortion (THD) Intermodulation Distortion (IMD) Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

Bandpass Filtering and Linear Distortion Equivalent Low-pass filter: Modeling a bandpass filter by using an equivalent low pass filter (complex impulse response) Bandpass filter H(f) = Y(f)/X(f) Input bandpass waveform Output bandpass waveform Impulse response of the bandpass filter Frequency response of the bandpass filter

Bandpass Filtering

Bandpass Filtering Theorem: The complex envelopes for the input, output, and impulse response of a bandpass filter are related by g1(t) – complex envelope of input k(t) – complex envelope of impulse response Also, Proof: Spectrum of the output is Spectra of bandpass waveforms are related to that of their complex enveloped But

Bandpass Filtering Thus, we see that Taking inverse fourier transform on both sides Any bandpass filter may be described and analyzed by using an equivalent low-pass filter. Equations for equivalent LPF are usually much less complicated than those for bandpass filters & so the equivalent LPF system model is very useful.

Linear Distortion For distortionless transmission of bandpass signals, the channel transfer function H(f) should satisfy the following requirements: The amplitude response is constant A- positive constant The derivative of the phase response is constant Tg – complex envelope delay Integrating the above equation, we get

Linear Distortion

Bandpass filter delays input info by Tg , whereas the carrier by Td Linear Distortion The channel transfer function is If the input to the bandpass channel is Then the output to the channel (considering the delay Tg due to ) is Using Phase response of a linear function of frequency Bandpass filter delays input info by Tg , whereas the carrier by Td Modulation on the carrier is delayed by Tg & carrier by Td

Received Signal Pulse The signal out of the transmitter Transmission medium (Channel) Carrier circuits Signal processing Information m input The signal out of the transmitter g(t) – Complex envelope of v(t) If the channel is LTI , then received signal + noise n(t) – Noise at the receiver input Signal + noise at the receiver input A – gain of the channel - carrier phase shift caused by the channel, Tg – channel group delay. Signal + noise at the receiver input

Nonlinear Distortion Amplifiers Non-linear Linear Circuits with memory and circuits with no memory Memory - Present output value ~ function of present input + previous input values - contain L & C No memory - Present output values ~ function only of its present input values. Circuits : linear + no memory – resistive ciruits linear + memory – RLC ciruits (Transfer function)

Nonlinear Distortion Assume no memory  Present output as a function of present input in ‘t’ domain If the amplifier is linear K- voltage gain of the amplifier In practice, amplifier output becomes saturated as the amplitude of the input signal is increased. output-to-input characteristic can be modelled as a Taylor’s expansion about vi =0 Where - output dc offset level - 1st order (linear) term - 2nd order (square law) term There will be nonlinear distortion on the output if K2,K3,… are non-zero.

2nd Harmonic Distortion with Nonlinear Distortion Harmonic Distortion associated with the amplifier output is determined by Applying a single sinusoidal test tone to the amplifier input Let the input test tone be represented by To the amplifier input Then the second-order output term is 2nd Harmonic Distortion with = In general, for a single-tone input, the output will be Vn – peak value of the output at the frequency nf0 The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by

Nonlinear Distortion Intermodulation distortion (IMD) of the amplifier is obtained by using a two-tone test signal : If the input (tone) signals are Then the second-order output term is IMD Harmonic distortion at 2f1 & 2f2 Second-order IMD is: