I. Previously on IET

I. Previously on IET

Complex Exponential Function Im-Axis ω Re-Axis

The Fourier Transform Representing functions in terms of complex exponentials with different frequencies

The Fourier Transform (Cosine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Sine Function) Im-Axis Im-Axis - -ω Re-Axis Re-Axis ω -

The Fourier Transform (Sine Function) Im-Axis Im-Axis + -ω Re-Axis Re-Axis ω +

The Fourier Transform (Sine Function) Im-Axis Im-Axis - Re-Axis Re-Axis -ω ω -

The Fourier Transform (Sine Function) Im-Axis -ω Im-Axis + Re-Axis Re-Axis ω +

Fourier Transform of Sinusoids Notes • A real value for the coefficients in the frequency domain means that the starting point for rotation is on the real axis • An Imaginary value for the coefficients in the frequency domain means that the starting point for rotation is on the imaginary axis j(1/2) 1/2 1/2 ω ω -ω -ω 0 0 -j(1/2)

Fourier Transform of Real Valued Functions Im-Axis Im-Axis Im-Axis A real-valued function in time implies that G(-f) = G*(f) ωn ω1 Re-Axis Re-Axis Re-Axis ω2 Im-Axis Im-Axis Im-Axis -ω2 Re-Axis Re-Axis Re-Axis -ω1 -ωn

Introduction to Digital Modulation: Pulse Code Modulation

Digital Communication Systems Source of Information User of Information Source Encoder Source Decoder Channel Encoder Channel Decoder Modulator De-Modulator Channel

Pulse Code Modulation • An analog message signal is converted to discrete form in both time and amplitude and then represented by a sequence of coded pulses

Pulse Code Modulation • Low Pass Filter • Confining the frequency content of the message signal • Sampling • To ensure perfect reconstruction of message signal at the receiver, the sampling rate must exceed twice the highest frequency component of the message signal (Sampling Theorem) • Quantization • Converting of analog samples to a set of discrete amplitudes • Encoding • Translating the discrete set of samples in a form suitable for digital transmission Source of continuous-time (i.e., analog) message signal Encoder PCM Signal Quantizer Sampler Low pass Filter Analog-to-Digital Converter

Sampling Process: Introductory Note Sampling of the signal spectrum in the frequency domain Periodic signal in the time domain By Duality Sampling of the signal in the time domain Making the spectrum of the signal periodic in the frequency domain

Sampling Process • Basic operation for digital communications • Converts an analog signal into a corresponding sequence of samples (usually spaced uniformly in time) • Questions • What should be the sampling rate? • Can we reconstruct the original signal after the sampling process?

Effect of Sampling on Frequency Content of Signals • Let assume that the frequency content of analog signal in the frequency domain is confined with W • Define TS as the sampling interval • Define fS as the sampling frequency m(t) M(f) t (sec.) f (Hz) W -W Representation of analog signal m(t) in time domain

fS>2W • By using a LPF with W<fcutoff<fS-W at the receiver, it is possible to reconstruct the original signal from received samples m(t) t (sec) TS=1/fS M(f) LPF -fS+W fS-W -W fS+W W 0 -fS-W -fS fS f (Hz) fcutoff

fS=2W • By using a LPF with fcutoff=W at the receiver, it is possible to reconstruct the original signal from received samples m(t) t (sec) TS=1/fS M(f) LPF -W 3W W 0 -3W -fS=-2W fS=2W f (Hz) fcutoff

fS<2W • It is no longer possible to reconstruct the original signal from received samples m(t) t (sec) TS=1/fS M(f) -W fS+W W fS-W 0 -fS-W -fS+W -fS fS f (Hz)

Sampling Theorem • Sampling Theorem states that • A band-limited signal of finite energy which has no frequency components higher than W Hz is completely described by specifying the values of the signal at instants of time separated by 1/2W seconds • A band-limited signal of finite energy which has no frequency components higher than W Hz may be completely recovered from knowledge of its samples taken at the rate of 2W samples per second • fS=2W is called the Nyquist Rate • tS=1/2W is called the Nyquist interval

Pulse Code Modulation Revisited • Let TQ represent the time interval between two consecutive quantized representation levels • Let TS represent the time interval between two consecutive m-ary encoded symbols Analog-to-Digital Converter Source of continuous-time (i.e., analog) message signal Encoder PCM Signal Quantizer Sampler Low pass Filter Representation Levels (vj) j=1,2,…,L Transmitting Filter m-ary Symbol Encoder PCM Signal sk(t) (uk) k=1,2,…,logmL

M-ary Encoder Examples 64 Quantized representation levels vk k=1,2,…,64 Sampling Rate = 1/TQ Binary Code uk k=1,2 Symbol Rate = 1/TS=6/TQ Binary Symbol Encoder 64 Quantized representation levels vk k=1,2,…,64 Sampling Rate = 1/TQ 4-ary Code uk k=1,2,3,4 Symbol Rate = 1/TS=3/TQ 4-ary Symbol Encoder

Transmitting Filter • The output from the m-ary encoder is still a logical variable rather than an actual signal • The transmitting filter converts the output of the m-ary encoder to a pulse signal • Example: • Square pulse transmitting filter Binary Code PCM Signal TS TS TS 1 t=TS t=0 t=3TS t=2TS t=4TS TS 0 t=TS t=0 t=3TS t=2TS +1 +1 +1 -1 4-ary Code PCM Signal 1 TS TS TS t=TS t=0 t=3TS t=2TS t=4TS TS 0 t=TS t=0 t=3TS t=2TS +1 +3 +3 -3

Optimal Receiving Filter yk(t) xk(t) sk(t) yk(TS) Transmitting Filter g(t) Receiving Filter h(t) + Sample at t=TS wk(t) At sampling Instant t=TS Optimality is maximized

Matched Filter Objective: • Design the optimal receiving filter to minimize the effects of AWGN • Matched Filter • h(t)=g(TS-t), i.e., H(f)=G*(f ) • Sample the output of receiving filter every TS yk(t) xk(t) sk(t) yk(TS) Transmitting Filter g(t) Receiving Filter h(t) PCM Signal + Sample at t=TS wk(t)

Matched Filter: Square Pulse Transmitting Filter Assume AWGN Noise wk(t) is negligible, binary symbols +1,+1,-1,+1 1 xk(t) Transmitting Filter g(t) TS 0 sk(t) t=TS t=0 t=3TS t=2TS t=4TS wk(t) + xk(t) yk(t) TS TS TS 1 Matched Filter g(TS-t) t=TS t=0 t=3TS t=2TS t=4TS TS 0 -TS yk(t) yk(iTS) Sample at t=TS TS TS TS yk(TS) t=TS t=0 t=3TS t=2TS t=4TS -TS

I. Previously on IET