DT Filters 2/19. Atousa Hajshirmohammadi, SFU

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1 1/19 ENSC380 Lecture 23 Objectives: Signals and Systems Fourier Analysis: Discrete Time Filters Analog Communication Systems Double Sideband, Sub-pressed Carrier Modulation (DSBSC) Amplitude Modulation (DSB with Carrier) Single Side Band Modulation (SSB) Spectrum Analyzer

2 DT Filters 2/19 Most of real world signals are CT by nature. However, with the advancements in the digital world, DT signals and systems are employed widely today. The advantages of working with DT signals and systems over CT are: DT signals can be stored indefinitely on magnetic media without degradation. DT systems can be built with much smaller components than CT systems, and are not sensitive to environmental changes, e.g. temperature. DT filters can have programmable parts which makes changing their parameters simple. We have already studied DT signals and systems. Here we will briefly revisit some DT systems from a filter point of view.

3 Discrete-Time Filters DT Lowpass Filter H( F)= e j2πf 3/19 []= 4 5 h n n u[ n]

4 Discrete-Time Filters 4/19 Comparison of DT lowpass filter impulse response with RC passive lowpass filter impulse response

5 Discrete-Time Filters 5/19 DT Lowpass Filter Frequency Response RC Lowpass Filter Frequency Response

6 Discrete-Time Filters 6/19 Moving-Average Filter H( F)= e jπnf drcl( F, N +1) h[]= n δ []+δ n n 1 [ ]+ δ[ n 2]+L+ δ n N N +1 [ ]

7 Discrete-Time Filters 7/19 Ideal DT Lowpass Filter Impulse Response Almost-Ideal DT Lowpass Filter Impulse Response Almost-Ideal DT Lowpass Filter Magnitude Frequency Response

8 Discrete-Time Filters 8/19 Almost-Ideal DT Lowpass Filter Magnitude Frequency Response in db

9 Communication Systems 9/19 A communication system is a transmission system that transmits a message (e.g. a TV program) from point A to point B, and a receiver system which receives and retrieves the original message. Voice or video signals are converted to electric voltages (electromagnetic wave) before transmission. These are low pass signals with BW usually not larger than a few khz. Modulation is used to transfer these signals to higher frequency bands for two main (independent) reasons: High frequency electromagnetic waves travel more efficiently and need much smaller transmitter and receiver antennas than low frequency signals. If several message signals are transmitted within their original low frequency band, they will obviously interfere with each other. Modulation is used to move each signal to a different frequency band. There are different modulation techniques, with different tradeoffs between bandwidth and energy efficiency, and practicality (simplicity) of the system. In the following we refer to the low pass message signal, x(t), as the modulating signal, and the high frequency sinusoid, cos(2πf c t) as carrier.

10 DSBSC Modulation 10/19 Double sideband subpressed-carrier method simply multiplies the message with the carrier: y(t) = x(t)cos(2πf c t)

11 Demodulation of DSBSC 11/19 Multiply the received signal by the same carrier: y(t)cos(2πf c t), and use a LPF to retrieve the original message x(t). This method is called synchronized demodulation because the receiver has to generate the exact frequency and phase of the carrier generated by the transmitter.

12 Amplitude Modulation (AM) 12/19 Synchronized demodulation is often very difficult in practice. Thus in commercial radio and television a different modulation technique is used called amplitude modulation. you text calls this method DSB Transmitted-Carrier (DSBTC). In this method the carrier signal is added to the DSBSC signal:

Demodulation of AM 13/19 Because of the added carrier, the envelope of the modulated signal is equal to the message signal: Demodulation of AM can be

13 Demodulation of AM 13/19 Because of the added carrier, the envelope of the modulated signal is equal to the message signal: Demodulation of AM can be done using a non-linear system called envelope detector: Advantage over DSBSC: No need to synchronization between transmitter and receiver Disadvantage:

14 Single Side Band (SSB) 14/19 It is possible to transmit only one of the sidebands (lower or upper) of a DSBSC signal without losing any part of the original message. This method is called SSB: Advantage: Disadvantage:

15 Demodulation of SSB 15/19 Very similar to demodulation of DSBSC (requires synchronization):

16 Quadrature Carrier Modulation 16/19 To save bandwidth, it is possible to transmit two different signals x 1 (t) and x 2 (t) by modulating two carriers which have the same frequency, but a phase difference of π/2.

17 DT Modulation 17/19 DT modulation is very similar to CT modulation. For example DT-DSBSC modulation simply is x[n]cos(2πf 0 n). Let s see how this modulation system works.

18 Spectrum Analyzer 18/19 Previously we saw a system that finds the power spectrum of a system: This systems is not very efficient because it requires a large number of very narrow-band BPFs.

19 Spectrum Analyzer (Cont.) 19/19 A better solution is the following system: The signal s components centered at frequency fc will be shifted to the zero-frequency after multiplying by the cosine function: The power of y(t), is in fact the power of the original signal x(t) within the frequency range f c ± f m. The cosine function generator sweeps a range of frequencies and for each frequency the power of the signal is found.

Fourier Transform Analysis of Signals and Systems

Fourier Transform Analysis of Signals and Systems Ideal Filters Filters separate what is desired from what is not desired In the signals and systems context a filter separates signals in one frequency