{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 500ch9 - EE603 Class Notes John Stensby Chapter 9 Commonly...

This preview shows pages 1–4. Sign up to view the full content.

EE603 Class Notes 09/04/09 John Stensby Updates at http://www.ece.uah.edu/courses/ee420-500/ 9-1 Chapter 9: Commonly Used Models: Narrow-Band Gaussian Noise and Shot Noise Narrow-band, wide-sense-stationary (WSS) Gaussian noise η (t) is used often as a noise model in communication systems. For example, η (t) might be the noise component in the output of a radio receiver intermediate frequency ( IF ) filter/amplifier. In these applications, sample functions of η (t) are expressed as η η ω η ω (t) (t)cos t (t)sin t = c c s c , (9-1) where ω c is termed the center frequency (for example, ω c could be the actual center frequency of the above-mentioned IF filter). The quantities η c (t) and η s (t) are termed the quadrature components (sometimes, η c (t) is known as the in-phase component and η s (t) is termed the quadrature component), and they are assumed to be real-valued. Narrow-band noise η (t) can be represented in terms of its envelope R(t) and phase φ (t). This representation is given as c (t) R(t)cos( t (t)) η = ω + φ , (9-2) where 2 2 c s 1 s c R(t) (t) (t) (t) tan ( (t) / (t)). η + η η η φ (9-3) Normally, it is assumed that R(t) 0 and – π < φ (t) π for all time. Note the initial assumptions placed on η (t). The assumptions of Gaussian and WSS behavior are easily understood. The narrow-band attribute of η (t) means that η c (t), η s (t), R(t) and φ (t) are low-pass processes; these low-pass processes vary slowly compared to cos ω c t; they

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
EE603 Class Notes 09/04/09 John Stensby Updates at http://www.ece.uah.edu/courses/ee420-500/ 9-2 are on a vastly different time scale from cos ω c t. Many periods of cos ω c t occur before there is notable change in η c (t), η s (t), R(t) or φ (t). A second interpretation can be given for the term narrow-band . This is accomplished in terms of the power spectrum of η (t), denoted as S η ( ω ). By the Wiener-Khinchine theorem, S η ( ω ) is the Fourier transform of R η ( τ ), the autocorrelation function for WSS η (t). Since η (t) is real valued, the spectral density S η ( ω ) satisfies ( ) 0 ( ) ( ). η η η ω ≥ ω = −ω S S S (9-4) Figure 9-1 depicts an example spectrum of a narrow-band process. The narrow-band attribute means that S η ( ω ) is zero except for a narrow band of frequencies around ±ω c ; process η (t) has a bandwidth (however it might be defined) that is small compared to the center frequency ω c . Power spectrum S η ( ω ) may, or may not , have ± ω c as axes of local symmetry . If ω c is an axis of local symmetry, then S S η η ω ω ω ω ( ) ( ) + = + c c (9-5) for 0 < ω < ω c , and the process is said to be a symmetrical band-pass process (Fig. 9-1 depicts a symmetrical band-pass process). It must be emphasized that the symmetry stated by the second of (9-4) is always true ( i.e., the power spectrum is even); however, the symmetry stated by (9-5) Rad/Sec ω S η ( ω ) - ω c ω c watts/Hz Fig. 9-1: Example spectrum of narrow-band noise.
EE603 Class Notes 09/04/09 John Stensby Updates at http://www.ece.uah.edu/courses/ee420-500/ 9-3 may, or may not, be true. As will be shown in what follows, the analysis of narrow-band noise is simplified if (9-5) is true.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 38

500ch9 - EE603 Class Notes John Stensby Chapter 9 Commonly...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online