On the role of waveform diversity in MIMO radar

Digital Signal Processing 23 (2013) 712–721

Contents lists available at SciVerse ScienceDirect

Digital Signal Processing www.elsevier.com/locate/dsp

On the role of waveform diversity in MIMO radar B. Friedlander Department of Electrical Engineering, University of California, Santa Cruz, CA 95064, United States

a r t i c l e

i n f o

Article history: Available online 20 December 2012 Keywords: Radar MIMO Ambiguity function Delay Doppler

a b s t r a c t MIMO radar employs multiple antennas to simultaneously transmit diverse waveforms, as well as multiple antennas to receive the radar returns. This paper studies the role of waveform diversity in MIMO radar as separate and distinct from the role of the multiple transmit antennas. This is done by comparing a MIMO radar system to a scanning phased array radar which uses the same transmit and receive arrays but only a single waveform. The performance characteristics of the two systems, in terms of the ambiguity function and the spatial response, are compared for single pulse operation as well as multi-pulse operation with coherent integration. Both element-space and beam-space systems are considered. © 2012 Elsevier Inc. All rights reserved.

1. Introduction In recent years there has been considerable interest in a class of radar systems called MIMO radar. While there does not seem to be universal agreement as to the precise definition of MIMO radar, the following is a commonly used description: “MIMO radar is characterized by using multiple antennas to simultaneously transmit diverse (possibly linearly independent) waveforms and by utilizing multiple antennas to receive the reflected signals.” [3] The literature on the subject addresses two distinct types of radar systems which we will refer to as MIMO radar with co-located antennas [1] and MIMO radar with widely separated antennas [2], which is also referred to as statistical MIMO. In this paper we consider only MIMO radars with co-located antennas and all subsequent mentions of MIMO radar refer to this type only. In discussions of MIMO radar in the literature its characteristics and performance are often contrasted with “conventional” radars employing a single transmit antenna and a single transmitted waveform, see e.g., [5–10]. These studies evaluate the combined effect of using more transmit antennas and diverse waveforms. This issue was also discussed in [11]. In this paper we attempt to separate the effects of waveform diversity from the effects of multiple transmit antennas in order to clarify their respective roles and gain some insights into the characteristics of MIMO radar. The approach taken here is to compare a MIMO radar with a radar system which is identical in every way, except that it uses only a single waveform. In other words, with a scanning phased

E-mail address: [email protected]. 1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.dsp.2012.12.008

array radar with the same number of transmit and receive antennas as the MIMO radar and the same system parameters such as power, bandwidth, pulse repetition frequency, etc. We consider the case where the phased array radar scans the transmit antenna elements, and the case where it scans a pre-formed set of transmit beams. We refer to these two modes as element-space and beam-space. Comparison is made to MIMO radar operating in element-space (different waveforms feeding different antennas) and beam-space (different waveforms feeding different beams), respectively. The structure of the paper is as follows. In Section 2 we present a model for the signal at the output of the matched filter receiver. This model allows us to treat both MIMO radars and phased array radars in a single common framework. In Section 3 we derive the delay-Doppler ambiguity function for MIMO radar. Section 4 presents a comparison of the structure of the waveforms in MIMO radars and scanning phased array radars in several configurations. In Section 5 we compare in some detail the delay-Doppler ambiguity functions of the two radar systems and their spatial processing capabilities. The last section provides some concluding remarks. 2. The signal model Consider a MIMO radar employing N T antennas at the transmitter and N R antennas at the receiver. We assume that the array aperture is sufficiently small so that the radar returns from a given scatterer are fully correlated across the array. In other words, all transmit/receive antenna pairs experience the same (complex) RCS. To simplify the presentation we assume that the two arrays are co-located, i.e. this is a mono-static radar [1]. The extension to the bi-static case is straightforward, as long as all antenna pairs still observe the same RCS. The arrays are characterized by the array manifolds: a R (θ) for the receive array and a T (θ) for the transmit

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

array, where θ is the direction relative to the array. We assume that the arrays and all the scatterers are in the same 2-D plane. The extension to the 3-D case is straightforward and all of the following results hold for that case as well. The baseband representation of the radar return from a single scatterer at direction θ0 and delay τ0 relative to the radar is given by

x(t ) = a R (θ0 )a T (θ0 ) T s(t − τ0 )h0 e j ω0 t + v(t )

Ignoring the phase term e − j (ω−ω0 )τ0 which can be absorbed into the unknown phase of the scatterer, the vector ambiguity function can be written in terms of the differences τ = τ − τ0 and ω = ω − ω0 , resulting in the more convenient form

T C(τ , ω) =

x(t )sn (t − τ )∗ e − j ωt dt

zn (τ , ω) =

Eq. (5) can now be written as



(10)





Z(τ , ω) T = C(τ , ω)a T (θ0 ) a R (θ0 ) T + W T

(11)

The vectorized version of the matched filter output is then given by



z(τ , ω) = vec Z(τ , ω) T



     = I ⊗ C(τ , ω) a R (θ0 ) ⊗ aT (θ0 ) + vec WT



w = vec W

zn (τ , ω) = a R (θ0 )a T (θ0 )

T ×

(12)

T



T =

s(t − τ0 )sn (t − τ )∗ e − j (ω−ω0 )t dt + wn (t )

v(t )sn (t − τ )∗ e − j (ω−ω0 )t dt

(3)

T T Rw = E 0

(4)

0

Stacking the matched filter output vectors for n = 1, . . . , N T side by side into an N R × N T matrix Z, we get

(13)



∗

v(t ) ⊗ s(t − τ )



 ∗ H

v ( u ) ⊗ s( u − τ )

 dt du

(14)

0

where R w is an N R N T × N R N T matrix. Using well-known Kronecker product identities we have

T T Rw = 0









E v(t )v(u ) H ⊗ s(t − τ )∗ s(u − τ ) T dt du

(15)

0

Note that

Z(τ , ω) = a R (θ0 )a T (θ0 ) T

×



v(t ) ⊗ s(t − τ )∗ e − j (ω−ω0 )t dt

0

where wn is the filtered N R × 1 noise vector,

T



Both z(τ , ω) and w are N R N T × 1 vectors. The covariance matrix of this noise vector is calculated as follows

T

0



s(t − τ0 )s(t − τ ) H e − j (ω−ω0 )t dt + W



E v(t )v(u ) H = σ 2 Iδ(t − u ) (5)

where the similarly stacked noise vectors are assembled into an N R × N T matrix

(16)

Therefore

0

T

+W

or

where zn (τ , ω) is an N R × 1 vector. Thus,

W=

T

Z(τ , ω) = a R (θ0 ) C(τ , ω)a T (θ0 )

(2)

0

wn =

(9)

where ⊗ denotes the Kronecker product, and

T

T

s(t − τ )∗ s(t ) T e − j ωt dt

0

(1)

where x(t ) is the N R × 1 vector of the receive array outputs at time t, s(t ) is an N T × 1 vector of the transmitted signals at the different transmit antennas, h0 is the amplitude of the scatterer and ω0 is the Doppler shift associated with it. Zero mean Gaussian noise v(t ) with variance σ 2 is added to the signal. We will consider for now a unit strength scatterer where h0 = 1. Note that a T (θ0 ) T s(t − τ0 ) is the radar illumination at the scatterer location. The received signal vector is processed by a bank of matched filters, each matched to one of the transmitted waveforms. The output of the n-th matched filter “tuned” to delay τ and Doppler ω is

713

T 2

Rw = σ I ⊗





s(t − τ )∗ s(t − τ ) T dt

(17)

0

v(t )s(t − τ ) H e − j (ω−ω0 )t dt

(6)

which can be written as

R w = σ 2 I ⊗ C(0, 0)

0

Let

T C(τ , ω, τ0 , ω0 ) =

s(t − τ )∗ s(t − τ0 ) T e − j (ω−ω0 )t dt

(7)

(18)

Note that if the signals are uncorrelated then C(0, 0) = I and the noise w is white. Next define by a(θ) the array manifold of the so-called virtual array

0

be the vector delay-Doppler ambiguity function of the transmitted signal s(t ). Because the radar waveform is assumed to be repeated every T seconds the ambiguity function can be re-written as

C(τ , ω, τ0 , ω0 ) = e − j (ω−ω0 )τ0

T × 0



∗

s t − (τ − τ0 ) s(t ) T e − j (ω−ω0 )t dt

(8)

a(θ) = a R (θ) ⊗ a T (θ)

(19)

Then the signal model can be summarized as follows. The N T N R × 1 vector of the output of the matched filter “tuned” to (τ , ω), in response to a unit strength scatterer at azimuth θ0 and at some delay and Doppler τ and ω relative to the matched filter delay-Doppler, is given by





z(τ , ω) = I ⊗ C(τ , ω) a(θ0 ) + w

(20)

714

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

where C(τ , ω) is defined in Eq. (9), and w is a noise vector whose covariance is defined in Eq. (18). Next consider a radar scene consisting of K scatterers with amplitudes hk , delays τk , and Dopplers ωk . The matched filter output is a superposition of outputs for individual scatterers

z(τ , ω) =

K





hk I ⊗ C(τk , ωk ) a(θk ) + w

(21)

k =1

In the following we will consider only the single scatterer case. The analysis we present is valid for the case of multiple scatterers as well. However, we limit our discussion to the case of a single scatterer for greater clarity and ease of explanation. 3. The MIMO delay-Doppler ambiguity function The matched filter outputs are combined by a beamformer weight vector b(θ) to produce the final output y of the receiver. Ignoring for now the additive noise the output is given by





y = b(θ) H z(τ , ω) = b(θ) H I ⊗ C(τ , ω) a(θ0 )

(22)

or



2

2 | y |2 = b R (θ) H a R (θ0 ) bT (θ)C(τ , ω)aT (θ0 )

(23)

where b(θ), b R (θ), b T (θ), are the beamformer weights,

b(θ) =

a(θ) , |a(θ)|

b R (θ) =

a R (θ) , |a R (θ)|

b T (θ) =

a T (θ) |aT (θ)|

(24)

Note that |b R (θ) H a R (θ0 )|2 is the power gain of the receive beamformer pointed at θ , in the direction θ0 . Similarly |b T (θ) H a T (θ0 )|2 is the power gain of the transmit beamformer pointed at θ , in the direction θ0 . If τ = 0 and ω = 0 then | y |2 will be the product of the transmit and receive beamformer gains, as expected. In this section we are interested in the delay-Doppler response and not the spatial response, so we will assume that the beamformer is correctly pointed at the scatterer, i.e. that θ = θ0 . Then



2

2 | y |2 = a R (θ) bT (θ) H C(τ , ω)a(θ)

(25)

or



2

2 | y |2 = a R (θ) aT (θ) AF(τ , ω; θ)

(26)

which is the conventional delay-Doppler ambiguity function for the composite signal sc (t ). Note that sc (t ) is the signal illuminating the scatterer at direction θ . Thus, the MIMO ambiguity function has the intuitive interpretation of being the ambiguity function of the composite waveform reflected from the scatterer. Note that the composite signal in (30) is a function of the direction θ of the scatterer. Thus the ambiguity function may change as a function of direction. In general, the ambiguity function of the composite signal will be different from the ambiguity functions of the component signals and its properties need to be investigated separately. A special case of interest is when the waveforms are random realizations of continuous-time bandlimited Gaussian white noise. Such waveforms are convenient surrogates for more carefully designed orthogonal waveforms and will be used throughout this paper. Because sc (t ) is a linear combination of these waveforms it will also be a realization of bandlimited Gaussian white noise with the same variance as the component waveforms (because the norm of the weight vector is unity). Therefore, the MIMO ambiguity function is statistically the same as the ambiguity functions of the component waveforms. Furthermore, note that if the elements of the transmit array are identical (i.e. have the same radiation pattern and orientation), then the entries of a T (θ) (and therefore all the entries of b T (θ)) will have the same magnitudes and differ only in phase. Because the phase does not change the statistical properties of complex circularly symmetric Gaussian noise, the composite signal sc (t ) will be a realization of bandlimited Gaussian white noise with the same variance as the component waveforms. Consequently, the shape of the MIMO ambiguity function is the same for all directions θ in this case. So far we discussed the element-space version of MIMO. In beam-space MIMO the transmitted waveforms are passed through a bank of beamformers. Let s(t ) be the signal vector at the antenna elements and s˜ (t ) be the set of orthogonal waveforms. Then

s(t ) = Ws˜ (t )

(32)

where W is an N T × N T matrix whose columns are the beamformer weights. In this case the composite signal is

sc (t ) = b T (θ) T W s˜ (t )





(33)

gn (θ)˜sn (t )

(34)



g(θ )

where







2

AF(τ , ω; θ) = b(θ) H I ⊗ C(τ , ω) b(θ)

or

(27) sc (t ) =

is the MIMO delay-Doppler ambiguity function. Because

T C(τ , ω) =

NT n =1

s(t − τ )∗ s(t ) T e − j ωt dt

(28)

0

we have

T

2



H ∗ T − j ωt

AF(τ , ω; θ) = b(θ) s(t − τ ) s(t ) b(θ)e dt (29)



0

So if we define the composite signal

sc (t ) = b T (θ) T s(t )

(30)

where { gn (θ)} are the gains of the beamformers in direction θ . If the beamformers are designed to be (approximately) orthogonal, all of the gains but one will be zero (or small), so that the composite signal will consist of only one component waveform. In this case the MIMO ambiguity function will equal the ambiguity function of that component signal. In other words, the transmit array generates a “fan” of beams. The scatterers within different beams are illuminated by different signals s˜n (t ), associated with a corresponding ambiguity function. The signals can be designed to make these ambiguity functions similar to each other or quite different from each other. 4. MIMO radar and phased array radar

we have

T

2



∗ − j ωt

AF(τ , ω; θ) = sc (t − τ ) sc (t )e dt



0

(31)

The signal model presented in the previous section describes the MIMO radar signal at the output of a bank of matched filter receivers. As we show next, the same model can be used to

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

Fig. 1. The structure of the transmitted waveforms for a MIMO radar (top figure) and a scanning phased array radar (bottom figure) for four pulse periods. Each triangle depicts a distinct pulse.

715

Fig. 2. The structure of the transmitted waveforms for a MIMO radar (top figure) and a scanning phased array radar (bottom figure) for a single pulse period. Each triangle depicts a distinct pulse.

describe a scanning phased array radar which uses a single waveform. We consider two different situations: (1) The case where the MIMO radar performs coherent integration over a block of N T pulses. In other words, the waveform s(t ) contains N T repetitions of a set of N T orthogonal waveforms. This is illustrated in the top part of Fig. 1 for N T = 4. Next consider an element scanning phased array radar transmitting a single waveform from different antennas at different pulse periods. This is illustrated in the bottom part of Fig. 1. (2) The case where the MIMO radar operates on a single pulse at a time. In this case the waveform s(t ) consists of a single instance of a set of N T orthogonal waveforms. This is illustrated in the top part of Fig. 2 for N T = 4. Next consider an element scanning phased array radar transmitting a single waveform from different antennas at different sub-periods during a single pulse. This is illustrated in the bottom part of Fig. 2. Note that both the MIMO radar and the scanning phased array radar are described by a common signal model. The only difference between them is in the choice of the waveforms s(t ). Furthermore, note that the set of waveforms used by the phased array radar are orthogonal, because the waveforms have non-overlapping time supports. Based on the definition quoted in the introduction we therefore conclude that an element scanning phased array radar is a MIMO radar with a particular choice of waveforms. In order to make a fair comparison both radar systems are assumed to use the same total energy over the coherent integration period. If we consider a transmitter with power P , the MIMO radar allocates a power of P / N T per waveform, while the phased array radar transmits each waveform at full power P . In the following it will be useful to consider an element scanning radar like the one described above which transmits a different orthogonal waveform at each step of the scan. This is illustrated in the bottom part of Fig. 3. In MIMO radar the orthogonal waveforms are transmitted simultaneously through the entire pulse period,

Fig. 3. The structure of the transmitted waveforms for a MIMO radar using simultaneous transmission (top figure), sequential transmission (bottom figure). Each triangle depicts a distinct pulse.

as depicted in the top part of Fig. 3. It is possible, instead, to transmit them sequentially in non-overlapping sub-periods. So far we assumed that the scanning radar switches from one antenna element to the next. Instead, we may consider an array employing a bank of N T beamformers, and have the scanning radar switch from one beamformer to the next. We will refer to this as the beam-space scanning radar, and compare it with beam-space MIMO radar.

716

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

Fig. 4. The delay-Doppler response (ambiguity function in dB scale) of a MIMO system (top) and its phased array counterpart (bottom) for N T pulses, element space.

5. Discussion

s(t ) =

Given the results developed in the previous sections we can now describe the key performance characteristics of the MIMO radar and a scanning phased array radar with identical radar parameters, differing only in the choice of waveforms. Some of the discussion below will be illustrated by numerical examples. These examples were produced for a radar system with the following parameters: radar bandwidth 10 MHz, pulse period 50 μs, the waveforms are white complex Gaussian noise filtered by a raised cosine filter with excess bandwidth parameter α = 0.05, N R = N T = 6 omni-directional antennas, element spacing λ/2 for the receiver array and 3λ for the transmitter array, carrier frequency 2 GHz. These parameters are not meant to represent any specific radar system and the particular values of the parameters do not affect the relative comparisons we are making here. 5.1. Delay-Doppler response The MIMO ambiguity function in (31) summarizes the delayDoppler response of the radar system. Next we examine this response for the four cases listed in the previous section. Let

f n (t ),

t ∈ [0, T ], n = 1, . . . , N T

(35)

be the set of orthogonal waveforms used by the MIMO radar during a single pulse of duration T , and let f(t ) be the vector whose elements are these waveforms. Let f 0 (t ) denote the single waveform used by the phased array radar during the same pulse period. 5.1.1. Coherent integration of N T pulses, element space In this case the signal s(t ) transmitted by the MIMO radar is the N T -fold repetition of the basic waveform f(t ),

N T −1

f(t − nT )

(36)

n =0

The corresponding composite signal sc (t ) is the repetition of the composite waveform f c (t ), where

f c (t ) = b T (θ) T f(t )

(37)

and

sc (t ) =

N T −1

f c (t − nT )

(38)

n =0

In case f n (t ) are realizations of bandlimited white noise, f c (t ) is just another realization of the same bandlimited white noise process. Because sc (t ) is the repetition of f c (t ), the central portion of its ambiguity function is identical to that of the ambiguity function of f c (t ), which is statistically the same as the ambiguity function of any one of the component waveforms f n (t ). The signal transmitted by the phased array radar will transmit the waveform f 0 (t ) in sequence from different antennas. Therefore the composite signal will be the N T -fold repetition of the waveform,

sc (t ) =

N T −1

f 0 (t − nT )

(39)

n =0

In case f 0 (t ) is a realization of bandlimited white noise, f 0 (t ) and f c (t ) are statistically the same, and therefore the composite signal sc (t ) of the phased array radar is statistically the same as that of the MIMO radar. It follows, therefore, that the ambiguity functions of the two radar systems are statistically the same. This is illustrated in Fig. 4 which depicts the MIMO ambiguity function on the top, and the phased array ambiguity function on the bottom.

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

717

Fig. 5. The delay-Doppler response (ambiguity function in dB scale) of a MIMO system (top) and its phased array counterpart (bottom) for N T pulses, beam space.

5.1.2. Coherent integration of N T pulses, beam space We assume that in beam-space mode a given scatterer will be illuminated by one beam only, say beam #1. Thus, the composite signal of the MIMO radar will be

sc (t ) =

N T −1

f 1 (t − nT )

The waveform f 0 (t ) of the phased array radar has duration T , which is N T times shorter than that of the MIMO waveform. The composite signal is then

sc (t ) = (40)

n =0

N T −1

f 0 (t − nT )

(43)

n =0

Because of the shorter duration of the illumination the Doppler response of the phased array radar will be N T times wider than that of the MIMO radar. The reduced illumination time also increases the sidelobe level of the ambiguity function. This is illustrated in Fig. 5 which depicts the MIMO ambiguity function on the top, and the phased array ambiguity function on the bottom.

The total duration N T T is the same as for MIMO, so the Doppler resolution is the same. However, because the underlying waveform has a duration of T , the unambiguous range is decreased by factor of N T . This can be seen in Fig. 6 which shows a cut through the ambiguity function along the delay axis at zero Doppler. The MIMO system shows a single peak at zero delay while the phased array shows multiple peaks separated by the pulse duration. Note also that the structure of the sidelobes of the ambiguity function is different for the two systems. This is illustrated in Fig. 7 which depicts the MIMO ambiguity function on the top, and the phased array ambiguity function on the bottom. Note that if the phased array radar were to use different noise waveforms on different pulses, the composite signal will be a noise waveform of length N T T , exactly like the composite waveform of the MIMO radar (rather than N T repetitions of a single waveform). Therefore the ambiguity functions of the two systems are statistically the same in this case. This is illustrated in Fig. 8.

5.1.3. Single pulse, element space Here we consider the case where the MIMO radar uses a single pulse in which case the composite signal is just the composite waveform

5.1.4. Single pulse, beam space As before, we assume that in beam-space mode the scatterer will be illuminated by one beam only, say beam #1. Since the MIMO radar uses a single pulse the composite signal is

sc (t ) = f c (t )

sc (t ) = f 1 (t )

The composite signal is statistically the same as in the previous case because f 1 (t ) and f c (t ) are both bandlimited white noise waveforms, so the ambiguity function is statistically the same as in element-space MIMO. Similarly, the phased array radar will illuminate the scatterer only during one pulse period, say the first, so that

 sc (t ) =

f 0 (t ) 0  t  T 0 T  t  ( N T − 1) T

(41)

(42)

In order to make a direct comparison with the multi-pulse cases we assume that the duration of the MIMO pulse is N T T (rather than T ).

(44)

In order to make a direct comparison with the multi-pulse cases we assume that the duration of the MIMO pulse is N T T (rather than T ).

718

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

Fig. 6. A cut through the ambiguity function along the delay axis at zero Doppler, for a MIMO system (top) and its phased array counterpart (bottom) for single pulse, element space.

Fig. 7. The delay-Doppler response (ambiguity function in dB scale) of a MIMO system (top) and its phased array counterpart (bottom) for single pulse, element space.

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

719

Fig. 8. The delay-Doppler response (ambiguity function in dB scale) of a MIMO system (top) and a phased array using different orthogonal waveforms on each scan (bottom) for single pulse, element space.

The composite signal of the phased array radar is

 sc (t ) =

f 0 (t ) 0  t  T 0 T  t  ( N T − 1) T

(45)

z = a(θ0 )

The duration during which the scatterer is illuminated is T , which is N T times shorter than the corresponding duration for the MIMO radar. This causes the Doppler response of the phased array radar to be N T times wider than that of the MIMO radar. There is also an increased sidelobe level of the ambiguity function. This is illustrated in Fig. 9 which depicts the MIMO ambiguity function on the top, and the phased array ambiguity function on the bottom. These four cases may be summarized as follows:

• N T pulses, element space – ambiguity functions of MIMO radar and scanning phased array radar are essentially the same. • N T pulses, beam space – scanning phased array radar has lower Doppler resolution and higher ambiguity function sidelobes. • Single pulse, element space – scanning phased array radar has smaller unambiguous range. • Single pulse, beam space – scanning phased array radar has lower Doppler resolution and higher ambiguity function sidelobes. 5.2. Spatial response To study the spatial response of the radar system we consider the output of the matched filter (20) when it is tuned to the correct delay and Doppler (i.e. τ = 0,  w = 0),





z = I ⊗ C(0, 0) a(θ0 )

As discussed earlier, both the MIMO radar and the scanning phased array radar employ orthogonal signals and therefore C(0, 0) = I, so that

(46)

(47)

In other words, the matched filter output for a unit strength scatterer is the virtual array manifold. This array manifold determines all the spatial characteristics of the radar system such as array gain, angular resolution, interference rejection capability and so on. Because the matched filter output (47) is the same for both radar systems, their spatial characteristics are the same. Consider for example the beam pattern given by (cf. Eq. (23) for the case where τ = 0,  w = 0)







2 2 P (θ) = b R (θ) H a R (θ0 ) b T (θ)C(0, 0)a T (θ0 )

(48)

If orthogonal signals are used C(0, 0) = I and







2 2 P (θ) = b R (θ) H a R (θ0 ) b T (θ)a T (θ0 )

(49)

In other words, the beam pattern is the product of the beam patterns of the transmit array and the receive array. Because both systems are assumed to have the same arrays, their beam patterns are the same. Fig. 10 depicts the beam patterns of the MIMO radar and the scanning phased array radar for the same N T = N R = 6 array considered in previous examples. Note the slight difference between the two beam patterns which is due to the fact that the MIMO signals are not perfectly orthogonal (because finite pieces of independent white noise have small but non-zero correlation values), while the phased array signals are perfectly orthogonal (because of the non-overlapping time supports). Had we used perfectly uncorrelated signals for the MIMO system the two beam patterns would have been identical.

720

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

Fig. 9. The delay-Doppler response (ambiguity function in dB scale) of a MIMO system (top) and its phased array counterpart (bottom) for single pulse, beam space.

Fig. 10. The azimuth response (beam pattern) of a MIMO system (top) and its phased array counterpart (bottom).

It should be noted that the current literature on MIMO radar contains confusing and misleading statements about the superior spatial capabilities of MIMO, see e.g. [1,3,4]. The claimed superior performance is often due to comparing a MIMO radar to

a radar equipped with a single transmit antenna, or not taking into account what can be achieved by scanning the phased array. Also, statements made in the literature give the impression that the N T N R -element virtual array is a novel characteristic of MIMO

B. Friedlander / Digital Signal Processing 23 (2013) 712–721

radar. It should be realized that any system which has a transmit array and a receive array can be represented by a virtual array. The virtual array is a direct consequence of the fact that a product of beampatterns (the transmit and receive beampatterns) corresponds to a convolution of the aperture illumination functions (which produces the virtual array). The MIMO radar is able to access the outputs of these virtual elements by using orthogonal signals transmitted from the different antennas. But so does the scanning phased array.

721

offers distinct advantages over scanning with a single waveform: improved Doppler resolution or larger unambiguous range can be achieved by waveform diversity. Acknowledgment This work was supported by the National Science Foundation under grant CCF-0725366. References

5.3. Output signal-to-noise ratio The two systems have the same signal-to-noise ratio (SNR) at the receiver output, both in the element-space and the beam-space versions. In element space, the phased array and the MIMO system illuminate the same radar scene throughout the integration time of N T pulses. During each pulse period the phased array uses a single antenna transmitting the full power P . The MIMO system uses N T antennas each transmitting power of P / N T . The total illuminating power is the same in both cases. In beam space, the phased array radar illuminates a given scatterer during one out of N T pulse periods at full power P . The MIMO radar illuminates the same scatterer during the entire N T pulse periods at a fractional power P / N T . The total integrated power is the same in both systems. We conclude that the output SNR of both systems is the same, and therefore their detection performance is the same, as long as the radar returns do not fluctuate during the integration time. 6. Conclusions In this paper we derived the ambiguity function for MIMO radar in a form which allows its interpretation as the usual (scalar) ambiguity function of the composite signal reflected from a scatterer. We then proceeded to study the role of waveform diversity by comparing a MIMO radar employing multiple orthogonal waveforms to a scanning phased array radar employing a single waveform. We have shown that a scanning phased array radar may, in fact, be considered to be a MIMO radar with a particular choice of waveforms. Not surprisingly therefore, the two system have the same fundamental capabilities, but may offer different tradeoffs. When the receiver performs coherent integration over N T pulses the performance of the two systems is essentially the same. When detection is based on a single pulse, using multiple waveforms

[1] J. Li, P. Stoica, MIMO radar with colocated antennas: Review of some recent work, IEEE Signal Process. Mag. 24 (5) (September 2007) 106–114. [2] A.M. Haimovich, R.S. Blum, L.J. Cimini, MIMO radar with widely separated antennas, IEEE Signal Process. Mag. 25 (1) (January 2008) 116–129. [3] J. Li, P. Stoica, MIMO Radar Signal Processing, John Wiley & Sons, 2009. [4] I. Bekkerman, J. Tabrikian, Target detection and localization using MIMO radars and sonars, IEEE Trans. Signal Process. 54 (10) (October 2006) 3873–3883. [5] L. Xu, J. Li, Iterative generalized-likelihood ratio test for MIMO radar, IEEE Trans. Signal Process. 55 (6) (June 2007) 2375–2385. [6] J. Li, L. Xu, P. Stoica, K.W. Forsythe, D.W. Bliss, Range compression and waveform optimization for MIMO radar: A Cramer–Rao bound based study, IEEE Trans. Signal Process. 56 (1) (January 2008) 218–232. [7] K.W. Forsythe, D.W. Bliss, Waveform correlation and optimization issues for MIMO radar, in: Proc. 39th Asilomar Conf. Signals, Systems and Computers, November 2005, pp. 1306–1310. [8] Jianguo Huang, Lijie Zhang, Yunshan Hou, Yong Jin, Modified subspace algorithms for DOA estimation using MIMO array, in: Proceedings of the 9th International Conference on Signal Processing (ICSP), 2008, pp. 195–198. [9] J. Li, P. Stoica, L. Xu, W. Roberts, On parameter identifiability of MIMO radar, IEEE Signal Process. Lett. 14 (2) (December 2007) 968–971. [10] Luzhou Xu, Jian Li, P. Stoica, Target detection and parameter estimation for MIMO radar systems, IEEE Trans. Aerosp. Electron. Syst. 44 (3) (2008) 927–939. [11] B. Friedlander, On the relationship between MIMO and SIMO radars, IEEE Trans. Signal Process. 57 (1) (January 2009) 394–398.

Benjamin Friedlander received the B.Sc. and the M.Sc. degrees in Electrical Engineering from the Technion – Israel Institute of Technology in 1968 and 1972, respectively, and the Ph.D. degree in Electrical Engineering and the M.Sc. degree in Statistics from Stanford University in 1976. From 1976 to 1985 he was at Systems Control Technology, Inc., Palo Alto, California. From November 1985 to July 1988 he was with Saxpy Computer Corporation, Sunnyvale, California. From 1989 to 1999 he was at the University of California at Davis. Since 1999 he is a professor of electrical engineering at the University of California at Santa Cruz. Dr. Friedlander is the recipient of the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASIP), the 1989 Technical Achievement Award of the Signal Processing Society, and the IEEE Third Millennium Medal.