Eeng Chapter4 Bandpass Signalling  Bandpass Filtering and Linear Distortion  Bandpass Sampling Theorem  Bandpass Dimensionality Theorem  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

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

Eeng Bandpass Filtering

Eeng Bandpass Filtering Theorem: g 1 (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 The complex envelopes for the input, output, and impulse response of a bandpass filter are related by

Eeng Bandpass Filtering Taking inverse fourier transform on both sides Thus, we see that  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.

Eeng 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 T g – complex envelope delay Integrating the above equation, we get Are these requirements sufficient for distortionless transmission?

Eeng Linear Distortion

Eeng 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 Modulation on the carrier is delayed by T g & carrier by T d Bandpass filter delays input info by T g, whereas the carrier by T d

Eeng Bandpass Sampling Theorem If a waveform has a non-zero spectrum only over the interval, where the transmission bandwidth B T is taken to be same as absolute BW, B T =f 2 -f 1, then the waveform may be reproduced by its sample values if the sampling rate is Theorem: Quadrature bandpass representation Let f c be center of the bandpass: x(t) and y(t) are absolutely bandlimited to B=B T /2 The sampling rate required to represent the baseband signal is Quadrature bandpass representation now becomes Where and samples are independent, two sample values are obtained for each value of n Overall sampling rate for v(t):

Eeng Bandpass Dimensionality Theorem  Assume that a bandpass waveform has a nonzero spectrum only over a frequency interval, where the transmission bandwidth B T is taken to be the absolute bandwidth given by B T =f 2 -f 1 and B T <<f 1.  The waveform may be completely specified over a T 0 -second interval by N Independent pieces of information. N is said to be the number of dimensions required to specify the information.

Eeng Received Signal Pulse The signal out of the transmitter Transmission medium (Channel) Carrier circuits Signal processing Carrier circuits Signal processing Information m input 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 - carrier phase shift caused by the channel, T g – channel group delay. A – gain of the channel

Eeng Amplifiers Non-linearLinear 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

Eeng Nonlinear Distortion Assume no memory  Present output as a function of present input in ‘t’ domain K- voltage gain of the amplifier If the amplifier is linear In practice, amplifier output becomes saturated as the amplitude of the input signal is increased. output-to-input characteristic (Taylor’s expansion): Where - output dc offset level - 1 st order (linear) term - 2 nd order (square law) term

Eeng Nonlinear Distortion Let the input test tone be represented by Harmonic Distortion associated with the amplifier output: Then the second-order output term is In general, for a single-tone input, the output will be V n – peak value of the output at the frequency nf 0 = To the amplifier input The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by 2 nd Harmonic Distortion with

Eeng Nonlinear Distortion Intermodulation distortion (IMD) of the amplifier: If the input (tone) signals are Then the second-order output term is IMD Harmonic distortion at 2f 1 & 2f 2 Second-order IMD is:

Eeng Nonlinear Distortion Third order term is The third term is The second term (cross-product) is Intermodulation terms at nonharmonic frequencies For bandpass amplifiers, where f 1 & f 2 are within the pasband, f 1 close to f 2, the distortion products at 2f 1 +f 2 and 2f 2 +f 1 ~ outside the passband Main Distortion Products