SKEU 2073 Section 01 FBME SIGNALS & SYSTEMS DR NOOR ASMAWATI BINTI SAMSURI P19a – Level 4 (FKE) asmawati@fke.utm.my

FOURIER TRANSFORM PART 3 DR. NAS 20132014-2

Outline Fourier Transform Application Filter Modulation/Demodulation Fourier Transform Part III

FILTER Filtering: useful part of a signal is separated from undesirable The range of frequencies that pass through a filter is called the pass-band, whereas the frequencies that do not pass through the filter is referred to as the stop-band. The most common types of filters are: High-pass filter, Low- pass filter, Band-pass filter, band stop filter

High-Pass Filter (HPF) Characterized by a stop-band that extends from to and a pass-band that extends from to infinity, where is the cutoff frequency of the filter.

Low-Pass Filter (LPF) Characterized by a pass-band that extends from to and a stop-band that extends from to infinity.

Example 1: Application of High Pass Filter 2 signals have been multiplied together that yields: Assume a certain application requires; , which can be obtained using HPF. Using Table C.2(a), the Fourier Transform of these signals are: Fourier Transform Part III

Example 1: (continue..) If signal g3(t) is passed through a HPF with transfer function: The filtering process can be mathematically written as: or Here, G3(ω) is the Fourier transform of the input signal to a system, and G4(ω) is the Fourier Transform of the output signal. Fourier Transform Part III

Example 1: (continue..) The frequency spectra for all the signals are given below: 5π 1200π -1200π ω |G3(ω)| -800π 800π 3π 1200π -1200π ω |G4(ω)| 1200π -1200π ω |H1(ω)| ωc -ωc Fourier Transform Part III

Example 2: Application of an ideal Low Pass Filter Using the same signal as Example 1, assume that an application requires a signal which can be obtained using HPF. Fourier Transform Part III

Example 2: (continue..) The frequency spectra for all the signals are given below: 800π -800π ω |H2(ω)| ωc -ωc 0.8 5π 1200π -1200π ω |G3(ω)| -800π 800π 4π 800π -800π ω |G5(ω)| Fourier Transform Part III

MODULATION An important technique in communication system to transmit low frequency signal. Without amplitude modulation, a long antenna is needed to received the signal. Therefore, a high frequency signal is applied to carry the transmitting signal.

Example 4: Amplitude Modulation Information signal to be transmitted, m(t) is multiplied to a carrier signal c(t) with higher frequency, ωc.  m(t) c(t)=cos ωct x(t)=m(t)cos ωct Mixer -ωc +ωB ωc -ω X(ω) -ωc -ωc -ωB ωc +ωB ωc -ωB Ac/2 -ωB ωB -ω M(ω) This modulated signal will be transmitted through a transmitter.

Amplitude Demodulation In receiver part, the signal will be processed to obtain back the information signal, m(t). The received signal x(t) will be multiplied with the carrier signal, cos(ωct) to produce a signal y(t) . To get back the information signal, the signal Y(ω), which is y(t) in frequency domain, has to pass through LPF with the given transfer function.

Example 5: Amplitude Demodulation The transfer function: |H(ω)| -ωB 2 ωB -2ωc +ωB 2ωc -ω |Y(ω)| -2ωc -2ωc -ωB 2ωc +ωB 2ωc -ωB Ac/4 Ac/2 ωB -ωB The spectrum: Spectrum of information signal: |M(ω)| -ωB ωB -ω A