Spectrum Of A Digital Signal

SPECTRUM OF DIGITAL SIGNALS

1 Objective The aim is to graphically portray the fundamental effects on digital signal within a frequency spectrum and to understand the effects of filters on digital signal. We also aim to understand the behaviour digital under frequency domain and time domain.

2 Task 1 RMS of a signal 50Hz sin wave Vpeak=2.275V

Vrms=1.61V measured with a digital oscilloscope

Figure1: sine wave 50Hz Vpeak to peak=4.56 measured with a digital oscilloscope.

Vrms=1.61V Multimeter reading Theoretical calculation 1.59Vrms

Vrms =

V peak 2

= 1.61V

The variation between the calculated and measured RMS values is very small, thus we approach almost ideal cases. In most instances there is a large variation between calculated values and measured values because practical components are defective.

SPECTRUM OF DIGITAL SIGNALS

RMS voltage with zero DC offset 50Hz block wave

Vpeak=2.51V

Vpeak-peak=5.08V

Figure2: 50Hz square wave

Multimeter reading {Vrms=2.72V} Theoretical computation {Vpeak=Vrms=2.51}, only for square waves.

3 Task 2 -3dB Filter Point Filter Design The required bandwidth is 100 kHz; a capacitor of 1.6nf was chosen to meet bandwidth specifications. R=

1 1 = = 994.7Ω , 1 k Ω was chosen. 2πfc 2π × 1.6 × 10 −9 × 100000

Vout

Figure4: RC low pass filter circuit

SPECTRUM OF DIGITAL SIGNALS

Table 1

freq(Hz) 10 20 31 40 50 60 70

Vin(V) 1.5 1.59 1.58 1.61 1.6 1.61 1.63

Vout(V) 1.53 1.59 1.58 1.61 1.6 1.61 1.63

Vout/Vin 1.02 1 1 1 1 1 1

log(freq)dB 1 1.30103 1.49136169 1.60205999 1.69897 1.77815125 1.84509804

80.9 90 100 110 120 130 140 150 160

1.66 1.63 1.65 1.67 1.68 1.62 1.61 1.66 1.66

1.66 1.64 1.65 1.67 1.69 1.62 1.62 1.66 1.67

1 1.006135 1 1 1.005952 1 1.006211 1 1.006024

1.90794852 1.95424251 2 2.04139269 2.07918125 2.11394335 2.14612804 2.17609126 2.20411998

170 180 190 200 251 300 350 400 450 500 600 700 800 900 1000 1500 2000 2500 3000 3500 4500 5500 6500 7500 8500 9500 10000 11000 12000 13000 14000

1.61 1.6 1.63 1.64 1.61 1.59 1.59 1.61 1.6 1.59 1.59 1.6 1.59 1.6 1.6 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59

1.62 1.6 1.64 1.65 1.62 1.6 1.6 1.62 1.61 1.6 1.6 1.61 1.6 1.61 1.61 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.59 1.59 1.59 1.59 1.58 1.58 1.58 1.57

1.006211 1 1.006135 1.006098 1.006211 1.006289 1.006289 1.006211 1.00625 1.006289 1.006289 1.00625 1.006289 1.00625 1.00625 1.006289 1.006289 1.006289 1.006289 1.006289 1.006289 1.006289 1.006289 1 1 1 1 0.993711 0.993711 0.993711 0.987421

2.23044892 2.25527251 2.2787536 2.30103 2.39967372 2.47712125 2.54406804 2.60205999 2.65321251 2.69897 2.77815125 2.84509804 2.90308999 2.95424251 3 3.17609126 3.30103 3.39794001 3.47712125 3.54406804 3.65321251 3.74036269 3.81291336 3.87506126 3.92941893 3.97772361 4 4.04139269 4.07918125 4.11394335 4.14612804

Katlego Mohlala

Electronics 4A01

Page 3

SPECTRUM OF DIGITAL SIGNALS 15000 16000 17000 18000 19000 20000 21000 22000 23000 24000 25500 26500 27500 28500 30000 35000 45000 50000 55000 60000 65000 70000 75000 80000 85000 90000 95000 100000 105000 110000 115000 125000 135000 145000 165000

1.59 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.57 1.58 1.57 1.58 1.57 1.57 1.57 1.56 1.56 1.56 1.56 1.55 1.55 1.55 1.55 1.54 1.54 1.51 1.54 1.54 1.54 1.54 1.53 1.53 1.53

1.57 1.57 1.56 1.56 1.56 1.55 1.55 1.54 1.54 1.53 1.52 1.52 1.52 1.51 1.5 1.48 1.42 1.4 1.36 1.33 1.29 1.26 1.22 1.19 1.16 1.13 1.1 1.05 1.05 1.02 0.992 0.941 0.889 0.856 0.778

0.987421 0.993671 0.987342 0.987342 0.987342 0.981013 0.981013 0.974684 0.974684 0.968354 0.968153 0.962025 0.968153 0.955696 0.955414 0.942675 0.904459 0.897436 0.871795 0.852564 0.826923 0.812903 0.787097 0.767742 0.748387 0.733766 0.714286 0.695364 0.681818 0.662338 0.644156 0.611039 0.581046 0.559477 0.508497

4.17609126 4.20411998 4.23044892 4.25527251 4.2787536 4.30103 4.32221929 4.34242268 4.36172784 4.38021124 4.40654018 4.42324587 4.43933269 4.45484486 4.47712125 4.54406804 4.65321251 4.69897 4.74036269 4.77815125 4.81291336 4.84509804 4.87506126 4.90308999 4.92941893 4.95424251 4.97772361 5 5.0211893 5.04139269 5.06069784 5.09691001 5.13033377 5.161368 5.21748394

Input

Output

Figure5: input-output waveforms of a sine wave through a filter.

SPECTRUM OF DIGITAL SIGNALS

Bode plot of trnsfer function vs dB frequency 6 5 4 Vout/Vin 3

-3dB Point Conner frequency

2 1 0 0

0.2

0.4 0.6 0.8 Log(frequency)(dB)

1

1.2

Figure6: bode plot Power comparison Assume a load resistance of 1Ω. 2 Vout 3. 2 V 2 4.12 = = = 10.24W P = = 16 .81W Pin = out R 1 R 1 The out power is about a sixth of the input power.

Task 3 Spectrum of a Digital Signal Time domain spectrum at a fundamental frequency of 500 Hz:

2.5V dc offset

Figure7: Time domain representation of square wave with 2.5V DC offset at 500 Hz.

SPECTRUM OF DIGITAL SIGNALS

Dc component at 0Hz st

1 Harmonic at 500 Hz

2nd Harmonic at 700 Hz

Figure8: frequency spectrum of 500 Hz square wave with harmonics placed odd multiples of the fundamental frequency.

Figure9: Frequency spectrum outline of 500 Hz square wave. With reference to figure8, the pulse harmonics are decreasing due to the fact that a Fourier transforms sums decreasing pulse amplitude with an increase in frequency. Knee frequency t rise1 = 1.656 × 10 −6 s , the rise time measured with digital oscilloscope?

k nee =

1 1 = = 192.73kHz π × Trise π × 1.656 × 10 − 6

SPECTRUM OF DIGITAL SIGNALS 4 Task 4 Filtering a Digital Signal Filter design Fundamental frequency was chosen as 500 Hz. Capacitance, c=4.7µf, we therefore calculate the required resistance. R=

1 1 = = 63.66Ω ≅ 64Ω , however a resistance of 63.66Ω was not 2πfc 2π (500)( 4.7 × 10 − 6 )

available during the practical therefore we used a 68Ω resistor. We then designed a low pass RC filter from these parameters.

a) Rise time tr

Rise time tr

Fall time tf

Fall time tf

Figure10: filter input output wave forms of the RC filter. Time domain spectra. b)

figure11

(a)

(b)

SPECTRUM OF DIGITAL SIGNALS Figure11: (a) depicts the FFT of the input waveform and (b) if the FFT of low pass filter output. Frequency domain spectra. With reference to figure11 (b) we can conclude that a low pass filter bypasses high frequency components and suppresses low frequency components and the dc value remains unchanged. Table 2

Rise time RMS voltage Fall time Vpeakpeak

Input 415.5μs

Output 708μs

1.74V 503.5μs

1.04V 776.6μs

4.48V

3.44V

k nee − frequency =

1 = 766 Hz , input signal. πt rise

k nee − frequency =

1 = 449.95 Hz , output signal. πt rise

Verification calculations

Filter rise time: t rise =

1 1 = = 636.6us πf π × 500

Measured rise time value=415.5µs. This could have been caused by the variation in the resistance value, we designed for 63.6Ω but we only had 68Ω available, however the difference is good enough for practical purposes. t rise1 = 1.656 × 10 −6 s 2 2 Output rise time= (t rise 1 + t rise 2 ) = 415.5us which is the same as the measured value.

Katlego Mohlala

Electronics 4A01

Page 8

SPECTRUM OF DIGITAL SIGNALS 5 Task 5 5.1 Effect of bandwidth on RC filters We cascade two filters as follows:

Figure12: series cascaded filters Filter1 has a bandwidth of 500Hz and filter2 has a bandwidth of 250Hz . a) Transfer function

H ( s) =

  Vout  1 1   =  Vin  1 + sR1C1  1 + sR2 C 2 

b)

Figure12: Input and output of cascaded filters. Same analogy in figure10 applies.

SPECTRUM OF DIGITAL SIGNALS Input having high frequency components

figure13: FFT

High frequency components fall outside the bandwidth, other frequencies are bypassed

(a) input

(b) output

c) Table 3

Rise time RMS voltage Fall time Vpeakpeak

Input 3.608μs

Output 760μs

2.09V 3.608μs

548mV 756μs

4.56V

1.92V

k nee − frequency =

1 1 = = 88kHz πt rise π × 3.608 × 10 −6

Verification Calculations Rise time 500Hz filter= 636.6µs Rise time 250Hz filter=

1

π × 250

= 1.27 ms

Conclusion The practical was succesful we managed to draw a graphical interface into the insight of filters affect digital signals and how signals are effect by variation in frequency. The -3dB point was observed and and we saw how the FFT functions relates to spectrum.

SPECTRUM OF DIGITAL SIGNALS

Katlego Mohlala

Electronics 4A01

Page 11