EELE445-14 - Montana State · PDF file · 2015-12-17EELE445-14 Lecture 12....

EELE445-14

Lecture 11Filter example,

Bandwidth definitions and BPSK example

Example: White noise through filter

• Find Sn(f) in Watts/Hz•The equivalent noise bandwidth of the filter•The output PSD, Sy(f)•The total output noise power in dBm and in Watts•The output rms noise voltage.•The output Vp-p (assume 6σ noise)•What would the filter BW have to be to reduce the noise

power by 12 dB?

RC LPFfc = 10 MHz0

Bandlimited Signals

Bandwidth Definitions1. Absolute bandwidth: B=f2-f1, when the spectrum is zero outside the interval

f1<f <f2 along the positive frequency axis. Example, white noise through an ideal bandpass filter

2. 3-dB bandwidth (or half-power bandwidth) is B=f2-f1 , where for frequencies insde the band f1<f <f2 , the power spectra, |S(f)|, fall now lower than ½ the maximum value of |S(f)|, and the maximum value occurs at a frequency inside the band.

)(fffffor

≤≤∀≥

−=:bandwidth) power-half (or bandwidth dB-3

Bandwidth Definitions

BPSK Signal

Figure 2–23 Spectrum of a BPSK signal.

BPSK Signal

Example 2–22 (continued)

BPSK SignalFigure 2–24 FCC-allowed envelope for B = 30 MHz.

SamplingPAM- Pulse Amplitude

ModulationPCM- Pulse Code Modulation

EELE445-14Lecture 12

EELE445-14

Lecture 12

Sampling

Sampling – The Cardinal SeriesSampling Theorem: Any physical waveform may be represented over the interval byt ∞≤≤∞−

where fs is a parameter assigned some convenient valuegreater than zero

Sampling – The Cardinal Series

( ) Bfs 2min = Nyquist Frequency

Sampling – The Cardinal Series

Sampling – The Cardinal SeriesFigure 2–17 Sampling theorem.

The voltage at t=nTs is only dueto the voltage at sample nTs and isnot dependent on any other sample.(Orthogonal in time!)

Figure 2–18 Impulse sampling.

Impulse Sampling

( ) [ ]L++++=

++⇒−= ∑∑∞

−∞=

)3cos(2)2cos(2)cos(211

)cos()(

tnDDnTtt

ωωωδ

πωϕ

ϕωδδ

Impulse Sampling

( ) )(∑∞

−∞=

sT nTtts

( ) [ ]

[ ] )(1)()(

)3cos(2)2cos(2)cos(21)()()(

∑∞

−∞=

++++==

twttwtws

Lωωωδ

Impulse Sampling- text

∑∑

−∞=

−−=

eqnTtnTw

nTttwtw

1712)()(

)()()(

Impulse SamplingThe spectrum of the impuse sampled signal is the spectrum of the unsampled signal that is repeated every fs Hz, where fs is the sampling frequency or rate (samples/sec). This is one of the basic principles of digital signal processing, DSP.

Note:This technique of impulse sampling is often used to

translate the spectrum of a signal to another frequency band that is centered on a harmonic of the sampling frequency, fs.

If fs>=2B, (see fig 2-18), the replicated spectra around each harmonic of fs do not overlap, and the original spectrum can be regenerated with an ideal LPF with a cutoff of fs/2.

Figure 2–18 Impulse sampling.

Impulse SamplingFigure 2–19 Undersampling and aliasing.

Natural SamplingGeneration of PAM with natural sampling (gating).

Natural Sampling

Duty cycle =1/3

Natural Sampling

df 3= at Null

Figure 3–4 Demodulation of a PAM signal (naturally sampled).

nth Nyquist region recovery

Figure 3–5 PAM signal with flat-top sampling.Impulse sample and hold

Figure 3–6 Spectrum of a PAM waveform with flat-top sampling.

Figure 3–7 PCM trasmission system.

Figure 3–8 Illustration of waveforms in a PCM system.

Figure 3–9 Compression characteristics (first quadrant shown).

Figure 3–9 Continued

Figure 3–10 Output SNR of 8-bit PCM systems with and without companding.

Quantization Noise,Analog to Digital Converter-A/D

EELE445Lecture 13

Figure 3–7 PCM trasmission system.

Quantization

Δx(t)

1maxmax

−==Δ nx xN

Quantization – Results in a Loss of Informationx(t)

Lost Information

After sampling, x(t)=xi

R∈ix

After Quantization:

R∈= x,x̂Q(x)

1maxmax

−==Δ nx xN

Quantization Noise

Quantization function:

Define the mean square distortion:

~))(()( 22

Δ≤−

andxxQxxq

Quantization

( ) noiseonquantizatithePN

Δ= ∫

where N=2n and xmax is ½ the A/D input range

Quantization Noise

So we can define the mean squared error distortion as:

The pdf of the error is uniformly distributed )(~ XQXX −=

−2Δ

SQNR – Signal to Quantization Noise Ratio

Example of SQNR for full scale sinewave done on board

Px may be found using:

The distortion, or “noise”, is therefore:

Where Px is the power of the input signal

SQNR – Linear Quantization

The SQNR decreases asThe input dynamic rangeincreases

U-Law Nonuniform PCMused to increase SQNR for given Px , xmax, and n

U=255 U.S

a-Law Nonuniform PCM

a=87.56 U.S

u-Law v.s. Linear Quantization

8 bitPx is signal powerRelative to full scale

Pulse Code Modulation, PCM, Advantagecompared with analog systems

• b is the number of bits

• γ is (S/N)basebandRelative to full scale

• PPM is pulse position modulation

EELE445-14 - Montana State · PDF file · 2015-12-17EELE445-14 Lecture 12....

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